4–30 |||| Evaluate the integral.

4.

5. 6.

8.

9. 10. y t 5

st 2 1 2 dty dx

sx 2 1 16

y sx 2 2 a 2

x 4 dxy 1

x 2s25 2 x 2 dx7.

y2

0 x 3sx 2 1 4 dxy2

s2

1

t 3st 2 2 1 dt

y2s3

0

x 3

s16 2 x 2 dx

1–3 |||| Evaluate the integral using the indicated trigonometric

substitution. Sketch and label the associated right triangle.

1. ;

2. ;

;

n n n n n n n n n n n n

x ­ 3 tan uy x 3

sx 2 1 9 dx3.

x ­ 3 sin uy x 3s9 2 x 2 dx

x ­ 3 sec uy 1

x 2sx 2 2 9 dx

|||| 7.3 Exercises

494 y y y y CHAPTER 7 TECHNIQUES OF INTEGRATION

Therefore

EXAMPLE 7 Evaluate .

SOLUTION We can transform the integrand into a function for which trigonometric substi-

tution is appropriate by first completing the square under the root sign:

This suggests that we make the substitution . Then and , so

We now substitute , giving and , so

­ 2s3 2 2x 2 x 2 2 sin21S x 1 1

2D 1 C

­ 2s4 2 u 2 2 sin21Su

2D 1 C

­ 22 cos u 2 u 1 C

­ y s2 sin u 2 1d du

y x

s3 2 2x 2 x 2 dx ­ y 2 sin u 2 1

2 cos u 2 cos u du

s4 2 u 2­ 2 cos udu ­ 2 cos u duu ­ 2 sin u

y x

s3 2 2x 2 x 2 dx ­ y u 2 1

s4 2 u 2 du

x ­ u 2 1du ­ dxu ­ x 1 1

­ 4 2 sx 1 1d2

3 2 2x 2 x 2­ 3 2 sx 2

1 2xd ­ 3 1 1 2 sx 21 2x 1 1d

y x

s3 2 2x 2 x 2 dx

­3

16 Fu 11

uG

1

1y2

­3

16 [(1

2 1 2) 2 s1 1 1d] ­3

32

y3s3y2

0

x 3

s4x 21 9d3y2

dx ­ 23

16 y1y2

1 1 2 u 2

u 2 du ­

3

16 y1y2

1 s1 2 u22 d du

|||| Figure 5 shows the graphs of the integrand

in Example 7 and its indefinite integral (with

). Which is which?C ­ 0

_4

_5

3

2

FIGURE 5

SECTION 7.3 TRIGONOMETRIC SUBSTITUTION y y y y 495

equation . Then is the sum of the area of the

triangle and the area of the region in the figure.]

; 36. Evaluate the integral

Graph the integrand and its indefinite integral on the same

screen and check that your answer is reasonable.

; 37. Use a graph to approximate the roots of the equation

. Then approximate the area bounded by

the curve and the line .

38. A charged rod of length produces an electric field at point

given by

where is the charge density per unit length on the rod and

is the free space permittivity (see the figure). Evaluate the inte-

gral to determine an expression for the electric field .

39. Find the area of the crescent-shaped region (called a lune)

bounded by arcs of circles with radii and . (See the figure.)

40. A water storage tank has the shape of a cylinder with diameter

10 ft. It is mounted so that the circular cross-sections are verti-

cal. If the depth of the water is 7 ft, what percentage of the

total capacity is being used?

41. A torus is generated by rotating the circle

about the -axis. Find the volume enclosed by the torus.x

x 21 sy 2 Rd2

­ r 2

R

r

Rr

0 x

y

L

P (a, b)

EsPd

«0l

EsPd ­ yL2a

2a

lb

4p«0sx 21 b 2 d3y2

dx

Psa, bdL

y ­ 2 2 xy ­ x 2s4 2 x 2

x 2s4 2 x 2­ 2 2 x

y dx

x 4sx 2 2 2

O x

y

RQ

¨

P

PQRPOQ

Ax 21 y 2

­ r 2

11. 12.

14.

15. 16.

18.

19. 20.

21.

23. 24.

25. 26.

27. 28.

29. 30.

(a) Use trigonometric substitution to show that

(b) Use the hyperbolic substitution to show that

These formulas are connected by Formula 3.9.3.

32. Evaluate

(a) by trigonometric substitution.

(b) by the hyperbolic substitution .

33. Find the average value of , .

34. Find the area of the region bounded by the hyperbola

and the line .

35. Prove the formula for the area of a sector of a circle

with radius and central angle . [Hint: Assume

and place the center of the circle at the origin so it has the

0 , u , py2ur

A ­1

2 r 2u

x ­ 39x 22 4y 2

­ 36

1 ø x ø 7f sxd ­ sx 2 2 1yx

x ­ a sinh t

y x 2

sx 21 a 2 d3y2

dx

y dx

sx 2 1 a 2­ sinh21S x

aD 1 C

x ­ a sinh t

y dx

sx 2 1 a 2­ ln(x 1 sx 2 1 a 2 ) 1 C

31.

n n n n n n n n n n n n

ypy2

0

cos t

s1 1 sin2t dty xs1 2 x 4 dx

y dx

s5 2 4x 2 x 2 d5y2y dx

sx 21 2x 1 2d2

y x 2

s4x 2 x 2 dxy 1

s9x 2 1 6x 2 8 dx

y dt

st 2 2 6t 1 13y s5 1 4x 2 x 2 dx

y1

0 sx 2 1 1 dx22.y2y3

0 x 3s4 2 9x 2 dx

y t

s25 2 t 2 dty s1 1 x 2

x dx

y dx

fsaxd22 b 2 g3y2y x

sx 2 2 7 dx17.

y dx

x 2s16x 2 2 9y x 2

sa 22 x 2 d3y2

dx

y du

us5 2 u 2y sx 2 2 9

x 3 dx13.

y1

0 xsx 2 1 4 dxy s1 2 4x 2 dx