Exercise - t Umathstat.sci.tu.ac.th/~archara/MA112/MA112-312/exercise112312-II-1.pdf · (b) 3x− 2y +z =1 4x+5y − 2z =4 (c) x− y +3z −2=0 2x+z =1 14. (a) 3x− 2y +z =4 6x−

MA112 Section 750001: Prepared by Dr. Archara Pacheenburawana 20

Exercise

Exercise 3.6

1− 4 Find an equation of the plane that passes through the point P and has the

vector n as normal.

1. P (2, 6, 1); n = 〈1, 4, 2〉

2. P (−1,−1, 2); n = 〈−1, 7, 6〉

3. P (1, 0, 0); n = 〈0, 0, 1〉

4. P (0, 0, 0); n = 〈2,−3,−4〉

5− 8 Find an equation of the plane indicated in the figure

5.

y1

z

1

x

1

6.

y1

z

1

x

1

7.

y1

z

1

x

1

8.

y1

z

1

x

1

9− 10 Find an equation of the plane that passes through the given point.

9. (−2, 1, 1), (0, 2, 3), and (1, 0,−1)

10. (3, 2, 1), (2, 1,−1), and (−1, 3, 2)

11− 12 Determine whether the planes are parallel, perpendicular, or neither.


13. (a) 2x− 8y − 6z − 2 = 0

−x+ 4y + 3z − 5 = 0

(b) 3x− 2y + z = 1

4x+ 5y − 2z = 4

(c) x− y + 3z − 2 = 0

2x+ z = 1

14. (a) 3x− 2y + z = 4

6x− 4y + 3z = 7

(b) y = 4x− 2z + 3

x = 14y + 1

2z

(c) x+ 4y + 7z = 3

5x− 3y + z = 0

13− 14 Determine whether the line and planes are parallel, perpendicular, or

neither.

13. (a) x = 4 + 2t, y = −t, z = −1− 4t;

3x+ 2y + z − 7 = 0

(b) x = t, y = 2t, z = 3t;

x− y + 2z = 5

(c) x = −1 + 2t, y = 4 + t, z = 1− t;

4x+ 2y − 2z = 7

14. (a) x = 3− t, y = 2 + t, z = 1− 3t;

2x+ 2y − 5 = 0

(b) x = 1− 2t, y = t, z = −t;

6x− 3y + 3z = 1

(c) x = t, y = 1− t, z = 2 + t;

x+ y + z = 1

15− 16 Determine whether the line and planes intersect; if so, find the

coordinates of the intersection.

15. (a) x = t, y = t, z = t;

3x− 2y + z − 57 = 0

(b) x = 2− t, y = 3 + t, z = t;

2x+ y + z = 1


16. (a) x = 3t, y = 5t, z = −t;

2x− y + z + 1 = 0

(b) x = 1 + t, y = −1 + 3t, z = 2 + 4t;

x− y + 4z = 7

17− 18 Find the acute angle of intersection of the planes.

17. x = 0 and 2x− y + z − 4 = 0

18. x+ 2y − 2z = 5 and 6x− 3y + 2z = 8

19− 28 Find an equation of the plane that satisfies the stated conditions.

19. The plane through the origin that is parallel to the plane 4x− 2y + 7z + 12 = 0.

20. The plane that contains the line x = −2+3t, y = 4+2t, z = 3−t and is perpendicular

to the plane x− 2y + z = 5.

21. The plane through the point (−1, 4, 2) that contains the line of intersection of the

planes 4x− y + z − 2 = 0 and 2x+ y − 2z − 3 = 0.

22. The plane through (−1, 4,−3) that is perpendicular to the line x− 2 = t, y+3 = 2t,

and z = −t.

23. The plane through (1, 2,−1) that is perpendicular to the line of intersection of the

planes 2x+ y + z = 2 and x+ 2y + z = 3.

24. The plane through the points P1(−2, 1, 4), P2(1, 0, 3) that is perpendicular to the

planes 4x− y + 3z = 2.

25. The plane through (−1, 2,−5) that is perpendicular to the planes 2x− y+ z = 1 and

x+ y − 2z = 3.

26. The plane that contains the point (2, 0, 3) and the line x = −1 + t, y = t, and

z = −4 + 2t.

27. The plane whose points are equidistant from (2,−1, 1) and (3, 1, 5).

28. The plane that contains the line x = 3t, y = 1 + t, z = 2t and is parallel to the

intersection of the planes y + z = −1 and 2x− y + z = 0.

29. Find parametric equations of the line through the point (5, 0,−2) that is parallel to

the planes x− 4y + 2z = 0 and 2x+ 3y − z + 1 = 0.

30. Let L be the line x = 3t+ 1, y = −5t, z = t.


(a) Show that L lies in the plane 2x+ y − z = 2.

(b) Show that L is parallel to the plane x + y + 2z = 0. Is the line above, below,

or on this plane?

31− 32 Find the distance between the point and the plane.

31. (1,−2, 3); 2x− 2y + z = 4

32. (0, 1, 5); 3x+ 6y − 2z − 5 = 0

33− 34 Find the distance between parallel planes.

33. (a) −2x+ y + z = 0

6x− 3y − 3z − 5 = 0

34. (b) x+ y + z = 1

x+ y + z = −1

35− 36 Find the distance between the given shew lines.

35. x = 1 + 7t, y = 3 + t, z = 5− 3t

x = 4− t, y = 6, z = 7 + 2t

36. x = 3− t, y = 4 + 4t, z = 1 + 2t

x = t, y = 3, z = 2t

37. Find an equation of the sphere with center (2, 1,−3) that is tangent to the plane

x− 3y + 2z = 4.

38. Locate the point of intersection of the plane 2x + y − z = 0 and the line through

(3, 1, 0) that is perpendicular to the plane.

Exercise 3.7

1. Identify the quadric surface as an ellipsoids, hyperboloids of one sheet, hyperboloids

of two sheet, elliptic cones, elliptic paraboloids, and hyperbolic paraboloids. State

the value of a, b, and c in each case.

(a) z =x2

4+

y2

9(b) z =

y2

25− x2

(c) x2 + y2 − z2 = 16 (d) x2 + y2 − z2 = 0

(e) 4z = x2 + 4y2 (f) z2 − x2 − y2 = 1


2. Find an equation of the trace, and state whether it is an ellipse, a parabola, or a

hyperbola

(a) 4x2 + y2 + z2 = 4; y = 1 (b) 4x2 + y2 + z2 = 4; x = 12

(c) 9x2 − y2 − z2 = 16; x = 2 (d) 9x2 − y2 − z2 = 16; z = 2

(e) z = 9x2 + 4y2; y = 2 (f) z = 9x2 + 4y2; z = 4

3− 8 Identify and sketch the quadric surface.

3. x2 +y2

4+

z2

9= 1 4.

x2

4+

y2

9− z2

16= 1

5. 4z2 = x2 + 4y2 6. 9z2 − 4y2 − 9x2 = 36

7. z = y2 − x2 8. 4z = x2 + 2y2

Exercise 3.8

1− 4 Convert from rectangular to cylindrical coordinates.

1. (4√3, 4,−4) 2. (−5, 5, 6)

3. (0, 2, 0) 4. (4,−4√3, 6)

5− 8 Convert from cylindrical to rectangular coordinates.

5. (4, π/6,−2) 6. (8, 3π/4,−2)

7. (5, 0, 4) 8. (7, π,−9)

9− 12 Convert from rectangular to spherical coordinates.

9. (1,√3,−2) 10. (1,−1,

√2)

11. (0, 3√3, 3) 12. (−5

√3, 5, 0)

13− 16 Convert from spherical to rectangular coordinates.

13. (5, π/6, π/4) 14. (7, 0, π/2)

15. (1, π, 0) 16. (2, 3π/2, π/2)

17− 20 Convert from cylindrical to spherical coordinates.


17. (√3, π/6, 3) 18. (1, π/4,−1)

19. (2, 3π/4, 0) 20. (6, 1,−2√3)

21− 24 Convert from spherical to cylindrical coordinates.

21. (5, π/4, 2π/3) 22. (1, 7π/6, π)

23. (3, 0, 0) 24. (4, π/6, π/2)

25− 28 An equation is given in cylindrical coordinates. Express the equation in

rectangular coordinates.

25. r = 3 26. z = r2

27. r = 4 sin θ 28. r2 + z2 = 1

29− 32 An equation is given in spherical coordinates. Express the equation in

rectangular coordinates.

29. ρ = 3 30. φ = π/4

31. ρ = 4 cosφ 32. ρ sinφ = 2 cos θ

33− 38 An equation of a surface is given in rectangular coordinates. Find an

equation of a surface in (a) cylindrical coordinates and (b) spherical coordinates.

33. z = 3 34. z = 3x3 + 3y2

35. x2 + y2 = 4 36. x2 + y2 + z2 = 9

37. 2x+ 3y + 4z = 1 38. x2 = 16− z2

Exercise 4.1

1− 4 Find the domain of r(t) and the value of r(t0).

1. r(t) = cos ti− 3tj ; t0 = π

2. r(t) = 〈√3t+ 1, t2〉 ; t0 = 1

3. r(t) = cosπti− ln tj +√t− 2k ; t0 = 3

4. r(t) = 〈2e−t, sin−1 t, ln(1− t)〉 ; t0 = 0


5− 8 Express the parametric equations as a single vector equation of the form

r = x(t)i + y(t)j or r = x(t)i + y(t)j+ z(t)k

5. x = 3 cos t, y = t+ sin t 6. x = t2 + 1, y = e−2t

7. x = 2t, y = 2 sin 3t, z = 5 cos 3t 8. x = t sin t, y = ln t, z = cos2 t

9− 12 Find the parametric equations that coorespond to the given vector equation.

9. r = 3t2i− 2j 10. r = sin2 ti+ (1− cos 2t)j

11. r = (2t− 1)i− 3√tj+ sin 3tk 12. r = te−ti− 5t2k

13− 18 Describe the graph of the equation.

13. r = (3− 2t)i+ 5tj 14. r = 2 sin 3ti− 2 cos 3tj

15. r = 2ti− 3j+ (1 + 3t)k 16. r = 3i+ 2 cos tj+ 2 sin tk

17. r = 2 cos ti− 3 sin tj+ k 18. r = −3i + (1− t2)j+ tk

19. (a) Find the slope of the line in 2-space that is represented by the vector equation

r = (1− 2t)i− (2− 3t)j

(b) Find the coordinates of the point where the line

r = (2 + t)i+ (1− 2t)j+ 3tk

intersects the xz-plane.

20. (a) Find the y-intercept of the line in 2-space that is represented by the vector equation

r = (3 + 2t)i+ 5tj

(b) Find the coordinates of the point where the line

r = ti + (1 + 2t)j− 3tk

intersects the plane 3x− y − z = 2.

21− 22 Sketch the line segment represented by the vector equation.

21. (a) r = (1− t)i + tj ; 0 ≤ t ≤ 1

(b) r = (1− t)(i+ j) + t(i− j) ; 0 ≤ t ≤ 1


22. (a) r = (1− t)(i + j) + tk ; 0 ≤ t ≤ 1

(b) r = (1− t)(i+ j + k) + t(i+ j) ; 0 ≤ t ≤ 1

23− 32 Sketch the graph of r(t) and show the direction of increasing t.

23. r(t) = 2i+ tj 24. r(t) = 〈3t− 4, 6t+ 2〉

25. r(t) = (1 + cos t)i + (3− sin t)j ; 0 ≤ t ≤ 2π

26. r(t) = 〈2 cos t, 5 sin t〉 ; 0 ≤ t ≤ 2π

27. r(t) =√t i + (2t+ 4)j 28. r(t) = 2 cos ti + 2 sin tj + tk

29. r(t) = 9 cos ti+ 4 sin tj + tk 30. r(t) = ti+ t2j+ 2k

31. r(t) = ti + tj+ sin tk ; 0 ≤ t ≤ 2π

Exercise 4.2

1− 2 Find the limit.

1. limt→+∞

⟨

t2 + 1

3t2 + 2,1

t

⟩

2. limt→0+

(√t i+

sin t

tj

)

3. limt→2

(

ti− 3j + t2k)

4. limt→1

⟨

3

t2,

ln t

t2 − 1, sin 2t

⟩

5− 6 Determine whether r(t) is continuous at t = 0. Explain your reasoning.

5. (a) r(t) = 3 sin ti− 2tj (b) r(t) = t2i+1

tj + tk

6. (a) r(t) = eti + tj+ csc tk (b) r(t) = 5i−√3t+ 1 j + e2tk

7− 8 Find r′(t).

7. r(t) = 4i− cos tj 8. r(t) = (tan−1 t)i+ t cos tj−√tk

9− 14 Find the vector r′(t0).

9. r(t) = 〈t, t2〉 ; t0 = 2 10. r(t) = t3i + t2j ; t0 = 1

11. r(t) = sec ti + tan tj ; t0 = 0 12. r(t) = 2 sin ti+ 3 cos tj ; t0 = π/6

13. r(t) = 2 sin ti + j+ 2 cos tk ; t0 = π/2 14. r(t) = cos ti + sin tj + tk ; t0 = π/4


15− 18 Find parametric equations of the line tangent to the graph of r(t) at the point

where t = t0.

15. r(t) = t2i+ (2− ln t)j ; t0 = 1

16. r(t) = e2ti− 2 cos 3tj ; t0 = 0

17. r(t) = 2 cosπti+ 2 sin πtj+ 3tk ; t0 =13

18. r(t) = ln ti+ e−tj + t3k ; t0 = 2

19− 22 Find a vector equation of the line tangent to the graph of r(t) at the point P0

on the curve.

19. r(t) = (2t− 1)i+√3t+ 4 j ; P0(−1, 2)

20. r(t) = 4 cos ti− 3tj ; P0(2,−π)

21. r(t) = t2i− 1t+1

j+ (4− t2)k ; P0 = (4, 1, 0)

22. Let r(t) = cos ti + sin tj + k. Find

(a) limt→0

(r(t)− r′(t)) (b) limt→0

(r(t)× r′(t))

(c) limt→0

(r(t) · r′(t))

23. Let r(t) = ti+ t2j+ t3k. Find limt→1

r(t) · r′(t)× r′′(t)

24− 25 Calculated

dx[r1(t) · r2(t)] and

d

dx[r1(t)× r2(t)]

first by differentiating the product directly and then by applying Formulas (13.11)

and (13.12).

24. r1(t) = 2ti + 3t2j + t3k, r2(t) = t4k

25. r1(t) = cos ti+ sin tj+ tk, r2(t) = i + tk

26− 31 Find the indefinite integral.

26.

∫

(3i+ 4tj) dt 27.

∫

(sin ti− cos tj) dt

28.

∫

(t sin ti + j) dt 29.

∫

〈tet, ln t〉 dt

30.

∫(

t2i− 2tj +1

tk

)

dt 31.

∫

⟨

e−t, et, 3t2⟩

dt


32− 37 Find the indefinite integral.

32.

∫ π/2

0

〈cos 2t, sin 2t〉 dt 33.

∫ 1

0

(t2i + t3j) dt

34.

∫ 2

0

‖ti+ t2j‖ dt 35.

∫ 3

−3

〈(3− t)3/2, (3 + t)3/2, 1〉 dt

36.

∫ 9

1

(

t1/2i+ t−1/2j)

dt 37.

∫ 1

0

(

e2ti+ e−tj + tk)

dt

Exercise 4.3

1− 4 Determine whether r(t) is a smooth function of the parameter t.

1. r(t) = t3i+ (3t2 − 2t)j+ t2k

2. r(t) = cos t2i+ sin t2j + e−tk

3. r(t) = te−ti+ (t2 − 2t)j+ cosπtk

4. r(t) = sin πti + (2t− ln t)j + (t2 − t)k

5− 8 Find the arc length of the parametric curve.

5. x = cos3 t, y = sin3 t, z = 2 ; 0 ≤ t ≤ π/2

6. x = 3 cos t, y = 3 sin t, z = 4t ; 0 ≤ t ≤ π

7. x = et, y = e−t, z =√2 t ; 0 ≤ t ≤ 1

8. x = 12t, y = 1

3(1− t)3/2, z = 1

3(1 + t)3/2 ; −1 ≤ t ≤ 1

9− 12 Find the arc length of the graph of r(t).

9. r(t) = t3i+ tj + 12

√6 t2k ; 1 ≤ t ≤ 3

10. r(t) = (4 + 3t)i+ (2− 2t)j+ (5 + t)k ; 3 ≤ t ≤ 4

11. r(t) = 3 cos ti+ 3 sin tj + tk ; 0 ≤ t ≤ 2π

12. r(t) = t2i+ (cos t+ t sin t)j+ (sin t− t cos t)k ; 0 ≤ t ≤ π

13− 16 Calculate dr/dτ by the chain rule.

13. r(t) = ti + t2j ; t = 4τ + 1

14. r(t) = 〈3 cos t, 3 sin t〉 ; t = πτ


15. r(t) = eti+ 4e−tj ; t = τ 2

16. r(t) = i+ 3t3/2j + tk ; t = 1/τ

17. (a) Find the arc length parametrization of the line

x = t, y = t

that has the same orientation as the given line and has reference point (0, 0).

(b) Find the arc length parametrization of the line

x = t, y = t, z = t

that has the same orientation as the given line and has reference point (0, 0, 0).

18. (a) Find the arc length parametrization of the line

x = −5 + 3t, y = 2t, z = 5 + t

that has the same direction as the given line and has reference point (−5, 0, 5).

(b) Use the parametric equations obtained in part (a) to find the point on the line that

is 10 units from the reference point in the direction of increasing parameter.

19− 21 Find an arc length parametrization of the curve that has the same

orientation as the given curve and for which the reference point corresponds

to t = 0.

19. r(t) = (4 + cos t)i + (5 + sin t)j ; 0 ≤ t ≤ 2π

20. r(t) = 13t3i+ 1

2t2j ; t ≥ 0

21. r(t) = sin eti + cos etj+√3etk ; t ≥ 0

Exercise 4.4

1− 9 Find T(t) and N(t) at the given point.

1. r(t) = (t2 − 1)i+ tj ; t = 1 2. r(t) = 12t2i+ 1

3t3j ; t = 1

3. r(t) = 5 cos ti+ 5 sin tj ; t = π/3 4. r(t) = ln ti + tj ; t = e

5. r(t) = 4 cos ti+ 4 sin tj+ tk ; t = π/2 6. r(t) = ti + 12t2j + 1

3t3k ; t = 0

7. x = et cos t, y = et sin t, z = et; t = 0 8. r(t) = cos ti + sin tj + k ; t = π/4

9. r(t) = eti+ et cos tj+ et sin tk ; t = 0


Exercise 5.1

1− 8 These exercises are concerned with functions of two variables.

1. Let f(x, y) = x2y + 1. Find

(a) f(2, 1) (b) f(1, 2) (c) f(0, 0)

(d) f(1,−3) (e) f(3a, a) (f) f(ab, a− b)

2. Let f(x, y) = x+ 3√xy. Find

(a) f(t, t2) (b) f(x, x2) (c) f(2y2, 4y)

3. Let f(x, y) = xy + 3. Find

(a) f(x+ y, x− y) (b) f(xy, 3x2y3)

4. Let g(x, y) = x sin y. Find

(a) g(x/y) (b) g(xy) (c) g(x− y)

5. Find F (g(x), h(x)) if F (x, y) = xexy, g(x) = x3, and h(y) = 3y + 1.

6. Find g(u(x, y), v(x, y)) if g(x, y) = y sin(x2y), u(x, y) = x2y3, and v(x, y) = πxy.

7. Let f(x, y) = x+ 3x2y2, x(t) = t2, and y(t) = t3. Find

(a) f(x(t), y(t) (b) f(x(0), y(0)) (c) f(x(2), y(2))

8. Let g(x, y) = ye−3x, x(t) = ln(t2 + 1), and y(t) =√t. Find g(x(t), y(t)).

9− 12 These exercises involve functions of three variables.

9. Let f(x, y, z) = xy2z3 + 3. Find

(a) f(2, 1, 2) (b) f(−3, 2, 1)

(c) f(0, 0, 0) (d) f(a, a, a)

10. Let f(x, y, z) = zxy + x. Find

(a) f(x+ y, x− y, x2) (b) f(xy, y/x, xz)

11. Find F (f(x), g(y), h(z)) if F (x, y, z) = yexyz, f(x) = x2, g(y) = y + 1, and h(z) =

z2.

12. Find g(u(x, y, z), v(x, y, z), w(x, y, z)) if g(x, y, z) = z sin xy, u(x, y, z) = x2z3,

v(x, y, z) = πxyz, and w(x, y, z) = xy/z.

13− 14 These exercises are concerned with functions of four or more variables.


13. (a) Let f(x, y, z, t) = x2y3√z + t. Find f(

√5, 2, π, 3π).

(b) Let f(x1, x2, . . . , xn) =n

∑

k=1

kxk. Find f(1, 1, . . . , 1).

14. (a) Let f(u, v, λ, φ) = eu+v cosλ tanφ. Find f(−2, 2, 0, π/4).

(b) Let f(x1, x2, . . . , xn) = x21 + x2

2 + · · ·+ x2n. Find f(1, 2, . . . , n).

15− 18 Sketch the domain of f . Use solid lines for portions of the boundary

included in the domain and dashed lines for portions not included.

15. f(x, y) = ln(1− x2 − y2) 16. f(x, y) =√

x2 + y2 − 4

17. f(x, y) =1

x− y218. f(x, y) = ln xy

19− 20 Describe the domain of f in words.

19. (a) f(x, y) = xe−√y+2 (b) f(x, y, z) =

√

25− x2 − y2 − z2

(c) f(x, y, z) = exyz

20. (a) f(x, y) =

√4− x2

y2 + 3(b) f(x, y) = ln(y − 2x)

(c) f(x, y, z) =xyz

x+ y + z

21− 30 Sketch the graph of f .

21. f(x, y) = 3 22. f(x, y) =√

9− x2 − y2

23. f(x, y) =√

x2 + y2 24. f(x, y) = x2 + y2

25. f(x, y) = x2 − y2 26. f(x, y) = 4− x2 − y2

27. f(x, y) =√

x2 + y2 + 1 28. f(x, y) =√

x2 + y2 − 1

29. f(x, y) = y + 1 30. f(x, y) = x2

Exercise 5.2

1− 6 Use limit laws and continuity properties to evaluate the limit.

1. lim(x,y)→(1,3)

(4xy2 − x) 2. lim(x,y)→(1/2,π)

(xy2 sin xy)

3. lim(x,y)→(−1,2)

xy3

x+ y4. lim

(x,y)→(1,−3)e2x−y2

5. lim(x,y)→(0,0)

ln(1 + x2y3) 6. lim(x,y)→(4,−2)

x 3√

y3 + 2x


7− 10 Show that the limit does not exist by considering the limits as

(x, y) → (0, 0) along the coordinate axes.

7. lim(x,y)→(0,0)

3

x2 + 2y28. lim

(x,y)→(0,0)

x+ y

2x2 + y2

9. lim(x,y)→(0,0)

x− y

x2 + y210. lim

(x,y)→(0,0)

cos xy

x2 + y2

11− 14 Evaluate the limit using the substitution z = x2 + y2 and observe that

z → 0+ if and only if (x, y) → (0, 0).

11. lim(x,y)→(0,0)

sin(x2 + y2)

x2 + y212. lim

(x,y)→(0,0)

1− cos(x2 + y2)

x2 + y2

13. lim(x,y)→(0,0)

e−1/(x2+y2) 14. lim(x,y)→(0,0)

e−1/√

x2+y2

√

x2 + y2

15− 22 Determine whether the limit exists. If so, find its value.

15. lim(x,y)→(0,0)

x4 − y4

x2 + y216. lim

(x,y)→(0,0)

x4 − 16y4

x2 + 4y2

17. lim(x,y)→(0,0)

xy

3x2 + 2y218. lim

(x,y)→(0,0)

1− x2 − y2

x2 + y2

19. lim(x,y,z)→(2,−1,2)

xz2√

x2 + y2 + z220. lim

(x,y,z)→(2,0,−1)ln(2x+ y − z)

21. lim(x,y,z)→(0,0,0)

sin(x2 + y2 + z2)√

x2 + y2 + z222. lim

(x,y,z)→(0,0,0)

sin√

x2 + y2 + z2

x2 + y2 + z2

Exercise 5.3

1. Let f(x, y) = 3x3y2. Find

(a) fx(x, y) (b) fy(x, y) (c) fx(1, y)

(d) fx(x, 1) (e) fy(1, y) (f) fy(x, 1)

(g) fx(1, 2) (h) fy(1, 2)

2. Let z = e2x sin y. Find

(a) ∂z/∂x (b) ∂z/∂y (c) ∂z/∂x|(0,y)

(d) ∂z/∂x|(x,0) (e) ∂z/∂y|(0,y) (f) ∂z/∂y|(x,0)

(g) ∂z/∂x|(ln 2,0) (h) ∂z/∂y|(ln 2,0)

3. Let f(x, y) =√3x+ 2y. Find

(a) Find the slope of the surface z = f(x, y) in the x-direction at the point (4, 2).


(b) Find the slope of the surface z = f(x, y) in the y-direction at the point (4, 2).

4. Let f(x, y) = xe−y + 5y. Find

(a) Find the slope of the surface z = f(x, y) in the x-direction at the point (3, 0).

(b) Find the slope of the surface z = f(x, y) in the y-direction at the point (3, 0).

5. Let z = sin(y2 − 4x). Find

(a) Find the rate of change of z with respect to x at the point (2, 1) with y held

fixed.

(b) Find the rate of change of z with respect to y at the point (2, 1) with x held

fixed.

6. Let z = (x+ y)−1. Find

(a) Find the rate of change of z with respect to x at the point (−2, 4) with y held

fixed.

(b) Find the rate of change of z with respect to y at the point (−2, 4) with x held

fixed.

7− 12 Find ∂z/∂x and ∂z/∂y.

7. z = 4ex2y3 8. z = cos(x5y4)

9. z = x3 ln(1 + xy−3/5) 10. z = exy sin 4y2

11. z =xy

x2 + y212. z =

x2y3√x+ y

13− 17 Find fx(x, y) and fy(x, y).

13. f(x, y) =√

5x5y − 7x3y 14. f(x, y) =x+ y

x− y

15. f(x, y) = y−3/2 tan−1(x/y) 16. f(x, y) = x3e−y + y3 sec√x

17. f(x, y) = (y2 tanx)−4/3

18− 21 Evaluate the indicated partial derivatives.

18. f(x, y) = 9− x2 − 7y3; fx(3, 1), fy(3, 1)

19. f(x, y) = x2yexy; ∂f/∂x(1, 1), ∂f/∂y(1, 1)

20. z =√

x2 + 4y2; ∂z/∂x(1, 2), ∂z/∂y(1, 2)


21. w = x2 cosxy; ∂w/∂x(

12, π

)

, ∂w/∂y(

12, π

)

22. Let f(x, y, z) = x2y4z3 + xy + z2 + 1. Find

(a) fx(x, y, z) (b) fy(x, y, z) (c) fz(x, y, z)

(d) fx(1, y, z) (e) fy(1, 2, z) (f) fz(1, 2, 3)

23. Let w = x2y cos z. Find

(a) ∂w/∂x(x, y, z) (b) ∂w/∂y(x, y, z) (c) ∂w/∂z(x, y, z)

(d) ∂w/∂x(2, y, z) (e) ∂w/∂y(2, 1, z) (f) ∂w/∂z(2, 1, 0)

24− 26 Find fx, fy, and fz.

24. f(x, y, z) = z ln(x2y cos z) 25. f(x, y, z) = y−3/2 sec

(

xz

y

)

26. f(x, y, z) = tan−1

(

1

xy2z3

)

27− 30 Find ∂w/∂x, ∂w/∂y, and ∂w/∂z.

27. w = yez sin xz 28. w =x2 − y2

y2 + z2

29. w =√

x2 + y2 + z2 30. w = y3e2x+3z

31. Let f(x, y, z) = y2exz. Find

(a) ∂f/∂x|(1,1,1) (b) ∂f/∂y|(1,1,1) (c) ∂f/∂z|(1,1,1)

32. Let w =√

x2 + 4y2 − z2. Find

(a) ∂w/∂x|(2,1,−1) (b) ∂w/∂y|(2,1,−1) (c) ∂w/∂z|(2,1,−1)

33. The volume V of a right circular cylinder is given by the formula V = πr2h, where

r is the radius and h is the height.

(a) Find the formula for the instantaneous rate of change of V with respect to r if

r changes and h remains constant.

(b) Find the formula for the instantaneous rate of change of V with respect to h if

h changes and r remains constant.

(c) Suppose that h has a constant value of 4 in, but r varies. Find the rate of

change of V with respect to r at the point where r = 6 in.

(d) Suppose that r has a constant value of 8 in, but h varies. Find the instantaneous

rate of change of V with respect to h at the point where h = 10 in.


34. The volume V of a right circular cone is given by

V =π

24d2√4s2 − d2

where s is the slant height and d is the diameter of the base.

(a) Find the formula for the instantaneous rate of change of V with respect to s if

d remains constant.

(b) Find the formula for the instantaneous rate of change of V with respect to d if

s remains constant.

(c) Suppose that d has a constant value of 16 cm, but s varies. Find the rate of

change of V with respect to s when s = 10 cm.

(d) Suppose that s has a constant value of 10 cm, but d varies. Find the rate of

change of V with respect to d when d = 16 cm.

35. According to the ideal gas law, the pressure, temperature, and volume of a gas are

related by P = kT/V , where k is a constant of proprotionality. Suppose that V is

measured in cubic inches (in3), T is measured in Kelvin (K), and that for a certain

gas the constant of proprotionality is k = 10 in-lb/K.

(a) Find the instantaneous rate of change of pressure with respect to temperature if

the temperature is 80 K and the volume remain fixed at 50 in3.

(b) Find the instantaneous rate of change of volume with respect to pressure if the

volume is 50 in3 and the temperature remain fixed at 80 K.

36. The length, width, and height of a rectangular box are ` = 5, w = 2, and h = 3,

respectively.

(a) Find the instantaneous rate of change of volume of the box with respect to the

length if w and h are held constant.

(b) Find the instantaneous rate of change of volume of the box with respect to the

width if ` and h are held constant.

(c) Find the instantaneous rate of change of volume of the box with respect to the

height if ` and w are held constant.

37. The area A of a triangle is given by A = 12ab sin θ, where a and b are the lengths of

two sides and θ is the angle between these sides. Suppose that a = 5, b = 10, and

θ = π/3.

(a) Find the rate at which A changes with respect to a if b and θ are held constant.

(b) Find the rate at which A changes with respect to θ if a and b are held constant.


(c) Find the rate at which b changes with respect to a if A and θ are held constant.

38. (a) By differentiating implicitly, find the slope of the hyperboloid x2 + y2− z2 = 1 in

the x-direction at the points (3, 4, 2√6) and (3, 4,−2

√6).

(b) Check the result in part (a) by solving for z and differentiating the resulting

functions directly.

39. (a) By differentiating implicitly, find the slope of the hyperboloid x2 + y2− z2 = 1 in

the y-direction at the points (3, 4, 2√6) and (3, 4,−2

√6).

(b) Check the result in part (a) by solving for z and differentiating the resulting

functions directly.

40− 43 Calculate ∂z/∂x and ∂z/∂y using implicit differentiation. Leave your

answers in terms of x, y, and z.

40. (x2 + y2 + z2)3/2 = 1 41. ln(2x2 + y − z3) = x

42. x2 + z sin xyz = 0 43. exy sinh z − z2x+ 1 = 0

44− 47 Find ∂w/∂x, ∂w/∂y, and ∂w/∂z using implicit differentiation. Leave

your answers in terms of x, y, z, and w.

44. (x2 + y2 + z2 + w2)3/2 = 4 45. ln(2x2 + y − z3 + 3w) = z

46. w2 + w sin xyz = 1 47. exy sinhw − z2w + 1 = 0

48. Let z =√x cos y. Find

(a) ∂2z/∂x2 (b) ∂2z/∂y2 (c) ∂2z/∂x∂y (d) ∂2z/∂y∂x

49. Let f(x, y) = 4x2 − 2y + 7x4y5. Find

(a) fxx (b) fyy (c) fxy (d) fyx

50. Given f(x, y) = x3y5 − 2x2y + x, find

(a) fxxy (b) fyxy (c) fyyy

51. Given z = (2x− y)5, find

(a)∂3z

∂y∂x∂y(b)

∂3z

∂x2∂y(c)

∂4z

∂x2∂y2

52. Given f(x, y) = y3e−5x, find

(a) fxxy(0, 1) (b) fxxx(0, 1) (c) fyyxx(0, 1)


53. Given w = ey cosx, find

(a)∂3w

∂y2∂x

∣

∣

∣

∣

(π/4,0)

(b)∂3w

∂x2∂y

∣

∣

∣

∣

(π/4,0)

54. Let f(x, y, z) = x3y5z7 + xy2 + y3z. Find

(a) fxy (b) fyz (c) fxz (d) fzz

(e) fzyy (f) fxxy (g) fzyx (h) fxxyz

55. Let w = (4x− 3y + 2z)5, find

(a)∂2w

∂x∂z(b)

∂3w

∂x∂y∂z(c)

∂4w

∂z2∂y∂x

Exercise 5.4

1− 6 Compute the differential dz or dw of the specified function.

1. z = 7x− 2y 2. z = exy

3. z = x3y2 4. z = 5x2y5 − 2x+ 4y + 7

5. z = tan−1 xy 6. z = sec2(x− 3y)

7. w = 8x− 3y + 4z 8. w = exyz

9. w = x3y2z 10. 4x2y3z7 − 3xy + z − 5

11. w = tan−1(xyz) 12. w =√x+

√y +

√z

13− 18 Use a total differential to approximate the change in the values of f from

P to Q. Compare your estimate with the actual change in f .

13. f(x, y) = x2 + 2xy − 4x; P (1, 2), Q(1.01, 2.04)

14. f(x, y) = x1/3y1/2; P (8, 9), Q(7.78, 9.03)

15. f(x, y) =x+ y

xy; P (−1,−2), Q(−1.02,−2.04)

16. f(x, y) = ln√1 + xy ; P (0, 2), Q(−0.09, 1.98)

17. f(x, y, z) = 2xy2z3; P (1,−1, 2), Q(0.99,−1.02, 2.02)

18. f(x, y, z) =xyz

x+ y + z; P (−1,−2, 4), Q(−1.04,−1.98, 3.97)

19− 26 (a) Find the local linear approximation L to the specified function f at

the designated point P . (b) Compare the error in approximating f by L at the

specified point Q with the distance between P and Q.


19. f(x, y) =1

√

x2 + y2; P (4, 3), Q(3.92, 3.01)

20. f(x, y) = x0.5y0.3; P (1, 1), Q(1.05, 0.97)

21. f(x, y) = x sin y; P (0, 0), Q(0.003, 0.004)

22. f(x, y) = ln xy; P (1, 2), Q(1.01, 2.02)

23. f(x, y, z) = xyz; P (1, 2, 3), Q(1.001, 2.002, 3.003)

24. f(x, y, z) =x+ y

y + z; P (−1, 1, 1), Q(−0.99, 0.99, 1.01)

25. f(x, y, z) = xeyz ; P (1,−1,−1), Q(0.99,−1.01,−0.99)

26. f(x, y, z) = ln(x+ yz); P (2, 1,−1), Q(2.02, 0.97,−1.01)

Exercise 5.5

1− 6 Use an appropriate form of the chain rule to find dz/dt.

1. z = 3x2y3; x = tt, y = t2 2. z = ln(2x2 + y); x =√t, y = t2/3

3. z = 3 cosx− sin xy; x = 1/t, y = 3t 4. z =√

1 + x− 2xy4; x = ln t, y = t

5. z = e1−xy; x = t1/3, y = t3

6− 9 Use an appropriate form of the chain rule to find dw/dt.

6. w = 5x2y3z4; x = t2, y = t3, z = t5

7. w = ln(3x2 − 2y + 4z3); x = t1/2, y = t2/3, z = t−2

8. w = 5 cosxy − sin xz; x = 1/t, y = t, z = t3

9. w =√

1 + x− 2yz4x; x = ln t, y = t, z = 4t

10− 15 Use an appropriate forms of the chain rule to find ∂z/∂u and ∂z/∂v.

10. z = 8x2y − 2x+ 3y; x = uv, y = u− v

11. z = x2 − y tan x; x = u/v, y = u2v2

12. z = x/y; x = 2 cosu, y = 3 sin v

13. z = 3x− 2y; x = y + v ln u, y = u2 − v ln v

14. z = ex2y; x =

√uv, y = 1/v


15. z = cosx sin y; x = u− v, y = u2 + v2

16− 23 Use an appropriate forms of the chain rule to find the derivatives.

16. Let T = x2y − xy3 + 2; x = r cos θ, y = r sin θ. Find ∂T/∂r and ∂T/∂θ.

17. Let R = e2s−t2 ; s = 3φ, t = φ1/2. Find dR/dφ.

18. Let t = u/v; u = x2 − y2, v = 4xy3. Find ∂t/∂x and ∂t/∂y.

19. Let w = rs/(r2 + s2); r = uv, s = u− 2v. Find ∂w/∂u and ∂w/∂v.

20. Let z = ln(x2 + 1), where x = r cos θ. Find ∂z/∂r and ∂z/∂θ.

21. Let u = rs2 ln t; r = x2, s = 4y + 1, t = xy3. Find ∂u/∂x and ∂u/∂y.

22. Let w = 4x2 + 4y2 + z2, x = ρ sinφ cos θ, y = ρ sin φ sin θ, z = ρ cos φ. Find

∂w/∂ρ, ∂w/∂φ, and ∂w/∂θ.

23. Let w = 3xy2z3, y = 3x2 + 2, z =√x− 1. Find dw/dx.

24. Use a chain rule to find the value ofdw

ds

∣

∣

∣

∣

s=1/4

if w = r2 − r tan θ; r =√s, θ = πs.

25. Use a chain rule to find the value of

∂f

∂u

∣

∣

∣

∣

u=1,v=−2

and∂f

∂v

∣

∣

∣

∣

u=1,v=−2

if f(x, y) = x2y2 − x+ 2y; x =√u, y = uv3.

26. Use a chain rule to find the value of

∂z

∂r

∣

∣

∣

∣

r=2,θ=π/6

and∂z

∂θ

∣

∣

∣

∣

r=2,θ=π/6

if z = xyex/y; x = r cos θ, y = r sin θ.

27. Use a chain rule to finddz

dt

∣

∣

∣

∣

t=3

if z = x2y; x = t2, y = t+ 7.

28− 31 Use Theorem 14.4 to find dy/dx and check your result using

implicit differentiation.

28. x2y3 + cos y = 0 29. x3 − 3xy2 + y3 = 5

30. exy + yey = 1 31. x−√xy + 3y = 4


32. Two straight roads intersect at right angles. Car A, moving on one of the roads, ap-

proaches the intersection at 25 mi/h and car B, moving on the other road, approaches

the intersection at 30 mi/h. At what rate is the distance between the cars changing

when A is 0.3 mile from the intersection and B is 0.4 mile from the intersection?

33. Use the ideal gas law P = kT/V with V in cubic inches (in3), T in kelvins (K),

and k = 10 in · lb/K to find the rate at which the temperature of a gas is changing

when the volume is 200 in3 and increasing at the rate of 4 in3/s, while the pressure

is 5 lb/in2 and decreasing at the rate of 1 lb/in2/s.

34. Two sides of a triangle have lengths a = 4 cm and b = 3 cm but are increasing at

the rate of 1 cm/s. If the area of the triangle remains constant, at what rate is the

angle θ between a and b changing when θ = π/6?

35. Two sides of a triangle have lengths a = 5 cm and b = 10 cm, and the included

angle is θ = π/3. If a is increasing at a rate of 2 cm/s, b is increasing at a rate of 1

cm/s, and θ remains constant, at what rate is the third side changing? Is it increasing

or decreasing? [Hint: Use the law of cosines.]

36. The length, width, and height of a rectangular box are increasing at rates of 1 in/s, 2

in/s, and 3 in/s, respectively.

(a) At what rate is the volume increasing when the length is 2 in, the width is 3 in,

and the height is 6 in?

(b) At what rate is the length of the diagonal increasing at that instant?

37. Consider the box in Exercise 36. At what rate is the surface area of the box increasing

at the given instant?

Exercise 5.6

1− 8 Find Duf at P .

1. f(x, y) = (1 + xy)3/2 ; P (3, 1) ; u =1√2i+

1√2j

2. f(x, y) = exy ; P (4, 0) ; u = −35i+ 4

5j

3. f(x, y) = ln(1 + x2 + y) ; P (0, 0) ; u = − 1√10

i− 3√10

j

4. f(x, y) =cx+ dy

x− y; P (3, 4) ; u = 4

5i+ 3

5j

5. f(x, y, z) = 4x5y2z3 ; P (2,−1, 1) ; u = 13i+ 2

3j− 2

3k


6. f(x, y, z) = yexz + z2 ; P (0, 2, 3) ; u = 27i− 3

7j+ 6

7k

7. f(x, y, z) = ln(x2 + 2y2 + 3z2) ; P (−1, 2, 4) ; u = − 313i− 4

13j− 12

13k

8. f(x, y, z) = sin xyz ; P(

12, 13, π

)

; u =1√3i− 1√

3j +

1√3k

9− 18 Find the directional derivative of f at P in the direction of a.

9. f(x, y) = 4x3y2 ; P (2, 1) ; a = 4i− 3j

10. f(x, y) = x2 − 3xy + 4y3 ; P (−2, 0) ; a = i+ 2j

11. f(x, y) = y2 ln x ; P (1, 4) ; a = 3i+ 3j

12. f(x, y) = ex cos y ; P (0, π/4) ; a = 5i− 2j

13. f(x, y) = tan−1(y/x) ; P (−2, 2) ; a = −i− j

14. f(x, y) = xey − yex ; P (0, 0) ; a = 5i− 2j

15. f(x, y, z) = x3z − yx2 + z2 ; P (2,−1, 1) ; a = 3i− j + 2k

16. f(x, y, z) = y −√x2 + z2 ; P (−3, 1, 4) ; a = 2i− 2j− k

17. f(x, y, z) =z − x

z + y; P (1, 0,−3) ; a = −6i+ 3j− 2k

18. f(x, y, z) = ex+y+3z ; P (−2, 2,−1) ; a = 20i− 4j+ 5k

19− 21 Find the directional derivative of f at P in the direction of a vector

making the counterclockwise angle θ with the positive x-axis.

19. f(x, y) =√xy ; P (1, 4) ; θ = π/3

20. f(x, y) =x− y

x+ y; P (−1,−2) ; θ = π/2

21. f(x, y) = tan(2x+ y) ; P (π/6, π/3) ; θ = 7π/4

22. Find the directional derivative of f(x, y) =x

x+ yat P (1, 0) in the direction of

Q(−1,−1).

23. Find the directional derivative of f(x, y) = e−x sec y at P (0, π/4) in the direction of

the origin.

24. Find the directional derivative of f(x, y) =√xy ey at P (1, 1) in the direction of the

negative y-axis.


25. Let

f(x, y) =y

x+ y

Find a unit vector u for which Duf(2, 3) = 0.

26. Find the directional derivative of

f(x, y, z) =y

x+ z

at P (2, 1,−1) in the direction form P to Q(−1, 2, 0).

27. Find the directional derivative of the function

f(x, y, z) = x3y2z5 − 2xz + yz + 3x

at P (−1,−2, 1) in the direction of the negative z-axis.

28− 31 Find ∇z or ∇w.

28. z = 4x− 8y 29. z = e−3y cos 4x

30. w = ln√

x2y2 + z2 31. w = e−5x sec x2yz

32− 35 Find the gradient of f at the indicate point.

32. f(x, y) = (x2 + xy)3 ; (−1,−1) 33. f(x, y) = (x2 + y2)−1/2 ; (3, 4)

34. f(x, y, z) = y ln(x+ y + z) ; (−3, 4, 0) 35. f(x, y, z) = y2z tan3 x ; (π/4,−3, 1)

Exercise 6.1

1− 12 Evaluate the iterated integral.

1.

∫ 1

0

∫ 2

0

(x+ 3) dy dx 2.

∫ 3

1

∫ 1

−1

(2x− 4y) dy dx

3.

∫ 4

2

∫ 1

0

x2y dx dy 4.

∫ 0

−2

∫ 2

−1

(x2 + y2) dx dy

5.

∫ ln 3

0

∫ ln 2

0

ex+y dy dx 6.

∫ 2

0

∫ 1

0

y sin x dy dx

7.

∫ 0

−1

∫ 5

2

dx dy 8.

∫ 6

4

∫ 7

−3

dy dx

9.

∫ 1

0

∫ 1

0

x

(xy + 1)2dy dx 10.

∫ π

π/2

∫ 2

1

x cos xy dy dx

11.

∫ ln 2

0

∫ 1

0

xyey2x dy dx 12.

∫ 4

3

∫ 2

1

1

(x+ y)2dy dx


13− 16 Evaluate the double integral over the rectangular region R.

13.

∫∫

R

4xy3 dA ; R = {(x, y) : −1 ≤ x ≤ 1, −2 ≤ y ≤ 2}

14.

∫∫

R

xy√

x2 + y2 + 1dA ; R = {(x, y) : 0 ≤ x ≤ 1, 0 ≤ y ≤ 1}

15.

∫∫

R

x√1− x2 dA ; R = {(x, y) : 0 ≤ x ≤ 1, 2 ≤ y ≤ 3}

16.

∫∫

R

(x sin y − y sin x) dA ; R = {(x, y) : 0 ≤ x ≤ π/2, 0 ≤ y ≤ π/3}

17− 20 Use a double integral to find the volume.

17. The volume under the plane z = 2x+ y and over the rectangle

R = {(x, y) : 3 ≤ x ≤ 5, 1 ≤ y ≤ 2}.

18. The volume under the surface z = 3x3 + 3x2y and over the rectangle

R = {(x, y) : 1 ≤ x ≤ 3, 0 ≤ y ≤ 2}.

19. The volume of the solid enclosed by the surface z = x2 and the planes x = 0, x =

2, y = 3, y = 0, and z = 0.

20. The volume in the first octant bounded by the coordinate planes, the plane y = 4,

and the plane (x/3) + (z/5) = 1.

21. Evaluate the integral by choosing a convenient order of integration:

∫∫

R

x cos(xy) cos2 πx dA ; R = [0, 12]× [0, π]

22. (a) Sketch the solid in the first octant that is enclosed by the planes x = 0, z = 0,

x = 5, z − y = 0, and z = −2y + 6.

(b) Find the volume of the solid by breaking it into two parts.


Exercise 6.2


1.

∫ 1

0

∫ x

x2

xy2 dy dx 2.

∫ 3/2

1

∫ 3−y

y

y dx dy

3.

∫ 3

0

∫

√9−y2

0

y dx dy 4.

∫ 1

1/4

∫ x

x2

√

x

ydx dy

5.

∫

√2π

√π

∫ x2

0

siny

xdy dx 6.

∫ 1

−1

∫ x

−x2

(x2 − y) dy dx

7.

∫ π

π/2

∫ x2

0

1

xcos

y

xdy dx 8.

∫ 1

0

∫ x

0

ex2

dy dx

9.

∫ 1

0

∫ x

0

y√

x2 − y2 dy dx 10.

∫ 2

1

∫ y2

0

ex/y2

dx dy

11. Let R be the region shown in the accompanying figure. Fill in the missing limits of

integration.

(a)

∫∫

R

f(x, y) dA =

∫

2

2

∫

2

2

f(x, y) dy dx

(b)

∫∫

R

f(x, y) dA =

∫

2

2

∫

2

2

f(x, y) dx dy

2x

y

R

y = x2


integration.

(a)

∫∫

R

f(x, y) dA =

∫

2

2

∫

2

2

f(x, y) dy dx

(b)

∫∫

R

f(x, y) dA =

∫

2

2

∫

2

2

f(x, y) dx dy


x

y

R

y =√x

y = x2


integration.

(a)

∫∫

R

f(x, y) dA =

∫ 2

1

∫

2

2

f(x, y) dy dx+

∫ 4

2

∫

2

2

f(x, y) dy dx

+

∫ 5

4

∫

2

2

f(x, y) dy dx

(b)

∫∫

R

f(x, y) dA =

∫

2

2

∫

2

2

f(x, y) dx dy

x

y

R

(2, 1) (4, 1)b b

bb(1, 3) (5, 3)


integration.

(a)

∫∫

R

f(x, y) dA =

∫

2

2

∫

2

2

f(x, y) dy dx

(b)

∫∫

R

f(x, y) dA =

∫

2

2

∫

2

2

f(x, y) dx dy

x

y

R

1−1

1

−1


15. Evaluate

∫∫

R

xy dA, where R is the region in

(a) Exercise 11 (b) Exercise 13

16. Evaluate

∫∫

R

(x+ y) dA, where R is the region in

(a) Exercise 12 (b) Exercise 14

17− 20 Evaluate the double integral in two ways using iterated integrals: (a) viewing R

as a type I region, and (b) viewing R as a type II region.

17.

∫∫

R

x2 dA; R is the region bounded by y = 16/x, y = x, and x = 8.

18.

∫∫

R

xy2 dA; R is the region bounded by y = 1, y = 2, x = 0, and y = x.

19.

∫∫

R

(3x− 2y) dA; R is the region enclosed by the circle x2 + y2 = 1.

20.

∫∫

R

y dA; R is the region in the first quadrant enclosed between the circle x2+y2 = 25

and the line x+ y = 5.

21− 26 Evaluate the double integral.

21.

∫∫

R

x(1+y2)−1/2 dA; R is the region in the first quadrant enclosed by y = x2, y = 4,

and x = 0.

22.

∫∫

R

x cos y dA; R is the triangular region bounded by the lines y = x, y = 0, and

x = π.

23.

∫∫

R

xy dA; R is the region enclosed by y =√x, y = 6− x, and y = 0.

24.

∫∫

R

x dA; R is the region enclosed by y = sin−1 x, x = 1/√2, and y = 0.

25.

∫∫

R

(x − 1) dA; R is the region in the first quadrant enclosed between y = x and

y = x3.


26.

∫∫

R

x2 dA; R is the region in the first quadrant enclosed by xy = 1, y = x, and

y = 2x.

27− 29 Use a double integration to find the area of the plane region enclosed by the

given curves.

27. y = sin x and y = cosx, for 0 ≤ x ≤ π/4.

28. y2 = −x and 3y − x = 4.

29. y2 = 9− x and y2 = 9− 9x.

30− 31 Use double integration to find the volume of the solid.

30.

x

y

z

3

6

4

3x+ 2y + 4z = 12

31.

x

y

z

2

2

2

x2 + z2 = 4

x2 + y2 = 4

32− 40 Use double integration to find the volume of the solid.

32. The solid bounded by the cylinder x2 + y2 = 9 and the planes z = 0 and z = 3− x.

33. The solid in the first octant bounded above by the paraboloid z = x2 + 3y2, below

by the plane z = 0, and laterally by y = x2 and y = x.

34. The solid bounded above by the paraboloid z = 9x2 + y2, below by the plane z = 0,

and laterally by the planes x = 0, y = 0, x = 3, and y = 2.


35. The solid enclosed by y2 = x, z = 0 and x+ z = 1.

36. The solid in the first octant bounded above by z = 9 − x2, below by z = 0, and

laterally by y2 = 3x.

37. The solid that is common to the cylinders x2 + y2 = 25 and x2 + z2 = 25.

38. The solid bounded above by the paraboloid z = x2 + y2, below by the xy-plane, and

laterally by the circular cylinder x2 + (y − 1)2 = 1.

39− 44 Express the integral as an equivalent integral with the order of integration

reversed.

39.

∫ 2

0

∫

√x

0

f(x, y) dy dx 40.

∫ 4

0

∫ 8

2y

f(x, y) dx dy

41.

∫ 2

0

∫ ey

1

f(x, y) dx dy 42.

∫ e

1

∫ lnx

0

f(x, y) dy dx

43.

∫ 1

0

∫ π/2

sin−1 y

f(x, y) dx dy 44.

∫ 1

0

∫

√y

y2f(x, y) dx dy

45− 48 Evaluate the integral by first reversing the order of integration.

45.

∫ 1

0

∫ 4

4x

e−y2 dy dx 46.

∫ 2

0

∫ 1

y/2

cos(x2) dx dy

47.

∫ 4

0

∫ 2

√y

ex3

dx dy 48.

∫ 3

1

∫ lnx

0

x dy dx

Exercise 6.3


1.

∫ π/2

0

∫ sin θ

0

r cos θ dr dθ 2.

∫ π

0

∫ 1+cos θ

0

r dr dθ

3.

∫ π/2

0

∫ a sin θ

0

r2 dr dθ 4.

∫ π/6

0

∫ cos 3θ

0

r dr dθ

5.

∫ π

0

∫ 1−sin θ

0

r2 cos θ dr dθ 6.

∫ π/2

0

∫ cos θ

0

r3 dr dθ

7− 10 Use a double integral in polar coordinates to find the area of the region

described.

7. The region enclosed by the cardioid r = 1− cos θ.


8. The region enclosed by the rose r = sin 2θ.

9. The region in the first quadrant bounded by r = 1 and r = sin 2θ, with π/4 ≤ θ ≤π/2.

10. The region inside the circle x2 + y2 = 4 and to the right of the line x = 1.

11− 12 Let R be the region described. Sketch the region R and fill in the

missing limits of integration.∫∫

R

f(r, θ) dA =

∫

2

2

∫

2

2

f(r, θ) dr dθ

11. The region inside the circle r = 4 sin θ and outside the circle r = 2.

12. The region inside the circle r = 1 and outside the cardioid r = 1 + cos θ.

13− 16 Use polar coordinates to evaluate the double integral.

13.

∫∫

R

e−(x2+y2) dA, where R is the region enclosed by the circle x2 + y2 = 1.

14.

∫∫

R

√

9− x2 − y2 dA, where R is the region in the first quadrant within the circle

x2 + y2 = 9.

15.

∫∫

R

1

1 + x2 + y2dA, where R is the sector in the first quadrant bounded by y = 0,

y = x, and x2 + y2 = 4.

16.

∫∫

R

2y dA, where R is the region in the first quadrant bounded above by the circle

(x− 1)2 + y2 = 1 and below by the line y = x.

17− 24 Evaluate the iterated integral by converting to polar coordinates.

17.

∫ 1

0

∫

√1−x2

0

(x2 + y2) dy dx 18.

∫ 2

−2

∫

√4−y2

−√4−yy

e−(x2+y2) dy dx

19.

∫ 2

0

∫

√2x−x2

0

√

x2 + y2 dy dx 20.

∫ 1

0

∫

√1−y2

0

cos(x2 + y2) dx dy

21.

∫ 1

0

∫

√y

y

√

x2 + y2 dx dy 22.

∫ a

0

∫

√a2−x2

0

1

(1 + x2 + y2)3/2dy dx (a > 0)

23.

∫ 4

0

∫

√25−x2

3

dy dx 24.

∫

√2

0

∫

√4−y2

y

1√

1 + x2 + y2dx dy


Exercise 6.4


1.

∫ 1

−1

∫ 2

0

∫ 1

0

(x2 + y2 + z2) dx dy dz 2.

∫ 1/2

1/3

∫ π

0

∫ 1

0

zx sin xy dz dy dx

3.

∫ 2

0

∫ y2

−1

∫ z

−1

yz dx dz dy 4.

∫ π/4

0

∫ 1

0

∫ x2

0

x cos y dz dx dy

5.

∫ 3

0

∫

√9−z2

0

∫ x

0

xy dy dx dz 6.

∫ 3

1

∫ x2

x

∫ ln z

0

xey dy dz dx

7.

∫ 2

0

∫

√4−x2

0

∫ 3−x2−y2

−5+x2+y2x dz dy dx 8.

∫ 2

1

∫ 2

z

∫

√3y

0

y

x2 + y2dx dy dz

9− 12 Evaluate the triple integral.

9.

∫∫∫

G

xy sin yz dV , where G is the rectangular box defined by the inequalities 0 ≤

x ≤ π, 0 ≤ y ≤ 1, 0 ≤ z ≤ π/6.

10.

∫∫∫

G

y dV , where G is the solid enclosed by the plane z = y, the xy-plane, and the

parabolic cylinder y = 1− x2.

11.

∫∫∫

G

xyz dV , where G is the solid in the first octant that is bounded by the parabolic

cylinder z = 2− x2 and the planes z = 0, y = x, and y = 0.

12.

∫∫∫

G

cos(z/y) dV , where G is the solid defined by the inequalities π/6 ≤ y ≤ π/2,

y ≤ x ≤ π/2, 0 ≤ z ≤ xy.

13− 15 Use a triple integral to find the volume of the solid.

13. The solid in the first octant bounded by the coordinate planes and the plane 3x +

6y + 4z = 12.

14. The solid bounded by the surface z =√y and the planes x + y = 0, x = 0, and

z = 0.

15. The solid bounded by the surface y = x2 and the planes y + z = 4 and z = 0.

16− 17 Set up (but do not evaluate) an iterated triple integral for the volume of the

solid enclosed between the given surfaces.


16. The elliptic cylinder x2 + 9y2 = 9 and the planes z = 0 and z = x+ 3.

17. The cylinders x2 + y2 = 1 and x2 + z2 = 1.

Exercise 7.1

1− 2 Confirm that φ is a potential function for F(r) on some region.

1. (a) φ(x, y) = tan−1 xy

F(x, y) =y

1 + x2y2i +

x

1 + x2y2j

(b) φ(x, y, z) = x2 − 3y2 + 4z2

F(x, y, z) = 2xi− 6yj+ 8zk

2. (a) φ(x, y) = 2y2 + 3x2y − xy3

F(x, y) = (6xy − y3)i+ (4y + 3x2 − 3xy2)j

(b) φ(x, y, z) = x sin z + y sin x+ z sin y

F(x, y, z) = (sin z + y cosx)i + (sin x+ z cos y)j+ (sin y + x cos z)k

3− 8 Find divF and curlF.

3. F(x, y, z) = x2i− 2j+ yzk 4. F(x, y, z) = xz3i + 2y4x2j+ 5z2yk

5. F(x, y, z) = 7y3z2i− 8x2z5j− 3xy4k 6. F(x, y, z) = exyi− cos yj+ sin2 zk

7. F(x, y, z) =1

√

x2 + y2 + z2(xi+ yj+ zk)

8. F(x, y, z) = ln xi + exyzj+ tan−1(z/x)k

9− 10 Find ∇ · (F×G).

9. F(x, y, z) = 2xi + j+ 4yk

G(x, y, z) = xi + yj− zk

10. F(x, y, z) = yzi+ xzj + xyk

G(x, y, z) = xyj+ xyzk

11− 12 Find ∇ · (∇× F).

11. F(x, y, z) = sin xi + cos(x− y)j+ zk 12. F(x, y, z) = exzi+ 3xeyj− eyzk

13− 14 Find ∇× (∇× F).

13. F(x, y, z) = xyj+ xyzk 14. F(x, y, z) = y2xi− 3yzj+ xyk


Exercise 7.2

1. Let C be the line segment from (0, 0) to (0, 1). In each part, evaluate the line integral

along C by inspection, and explain your reasoning.

(a)

∫

C

ds (b)

∫

C

sin xy dy

2. Let C be the line segment from (0, 2) to (0, 4). In each part, evaluate the line integral

along C by inspection, and explain your reasoning.

(a)

∫

C

ds (b)

∫

C

exy dx

3. Let C be the curve represented by the equations

x = 2t, y = 3t2 (0 ≤ t ≤ 1)

In each part, evaluate the line integral along C.

(a)

∫

C

(x− y) ds (b)

∫

C

(x− y) dx

(c)

∫

C

(x− y) dy

4. Let C be the curve represented by the equations

x = t, y = 3t2, z = 6t3 (0 ≤ t ≤ 1)

In each part, evaluate the line integral along C.

(a)

∫

C

xyz2 ds (b)

∫

C

xyz2 dx

(c)

∫

C

xyz2 dy (d)

∫

C

xyz2 dz

5. In each part, evaluate the integral∫

C

(3x+ 2y) dx+ (2x− y) dy

along the stated curve.

(a) The line segment from (0, 0) to (1, 1).

(b) The parabolic arc y = x2 from (0, 0) to (1, 1).

(c) The curve y = sin(πx/2) from (0, 0) to (1, 1).

(d) The curve x = y3 from (0, 0) to (1, 1).

6. In each part, evaluate the integral∫

y dx+ z dy − x dz

along the stated curve.


(a) The line segment from (0, 0, 0) to (1, 1, 1).

(b) The twisted cubic x = t, y = t2, z = t3 from (0, 0, 0) to (1, 1, 1).

(c) The helix x = cos pit, y = sin πt, z = t from (1, 0, 0) to (−1, 0, 1).

7− 14 Evaluate the line integral along the curve C.

7.

∫

C

(x+ 2y) dx+ (x− y) dy

C : x = 2 cos t, y = 4 sin t (0 ≤ t ≤ π/4)

8.

∫

C

(x2 − y2) dx+ x dy

C : x = t2/3, y = t (−1 ≤ t ≤ 1)

9.

∫

C

−y dx+ x dy

C : y2 = 3x from (3, 3) to (0, 0).

10.

∫

C

(y − x) dx+ x2y dy

C : y2 = x3 from (1,−1) to (1, 1).

11.

∫

C

(x2 + y2) dx− x dy

C : x2 + y2 = 1, counterclockwise from (1, 0) to (0, 1).

12.

∫

C

(y − x) dx+ xy dy

C : the line segment from (3, 4) to (2, 1).

13.

∫

C

yz dx− xz dy + xy dz

C : x = et, y = e3t, z = e−t (0 ≤ t ≤ 1)

14.

∫

C

x2 dx+ xy dy + z2 dz

C : x = sin t, y = cos t, z = t2 (0 ≤ t ≤ π/2)

15− 16 Evaluate∫

Cy dx− x dy along the curve C shown in the figure.

15. (a)

x

y

(1, 0)

(0, 1)


(b)

x

y

(1, 0)

(1, 1)

16. (a)

x

y

(2, 0)

(1, 1)

(b)

x

y

(0, 5)

(5, 0)(−5, 0)

17− 20 Evaluate∫

CF · dr along the curve C.

17. F(x, y) = x2i+ xyj

C : r(t) = 2 cos ti+ 2 sin tj (0 ≤ t ≤ π)

18. F(x, y) = x2yi+ 4j

C : r(t) = eti+ e−tj (0 ≤ t ≤ 1)

19. F(x, y) = (x2 + y2)−3/2(xi + yj)

C : r(t) = et sin ti+ et cos tj (0 ≤ t ≤ 1)

20. F(x, y, z) = zi + xj + yk

C : r(t) = sin ti + 3 sin tj+ sin2 tk (0 ≤ t ≤ π/2)

21− 24 Find the work done by the force field F on a particle that moves along

the curve C.


21. F(x, y) = xyi+ x2j

C : x = y2 from (0, 0) to (1, 1).

22. F(x, y) = (x2 + xy)i+ (y − x2y)j

C : x = t, y = 1/t (1 ≤ t ≤ 3)

23. F(x, y, z) = xyi+ yzj+ xzk

C : r(t) = ti + t2j + t3k (0 ≤ t ≤ 1)

24. F(x, y, z) = (x+ y)i+ xyj− z2k

C : along line segments from (0, 0, 0) to (1, 3, 1) to (2,−1, 4).

25− 26 Find the work done by the force field

F(x, y) =1

x2 + y2i+

4

x2 + y2j

on a particle that moves along the curve C show in the figure.

25.

x

y

(0, 4)

(4, 0)

26.

x

y

(6, 3)

b

b

Documents

Exercise - t Umathstat.sci.tu.ac.th/~archara/MA112/MA112-312/exercise112312-II-1.pdf · (b) 3x− 2y +z =1 4x+5y − 2z =4 (c) x− y +3z −2=0 2x+z =1 14. (a) 3x− 2y +z =4 6x−