Day 1: (Review) 5.1 & 5.2 & 5.3 velocity distance & The de nite … · 2019-09-04 · Fall 2019 Math 116: Calculus II C. Ashley Day 1: (Review) 5.1 & 5.2 & 5.3 velocity !distance

Fall 2019 Math 116: Calculus II C. Ashley

Day 1: (Review) 5.1 & 5.2 & 5.3velocity → distance & The definite Integral & FTC

September 4, 2019

5.1: How Do We Measure Speed?

1. The figure below shows the velocity of a car for 0 ≤ t ≤ 12 and the rectangles used to estimate ofthe distance traveled.

(a) Do the rectangles represent a left or a right sum?

(b) Do the rectangles lead to an upper or a lower estimate?

(c) What is the value of n?

(d) What is the value of ∆t?

(e) Give an approximate value for the estimate.

5.1 HOW DO WE MEASURE DISTANCE TRAVELED? 277

v = f(t)

v

a = t0 t1 t2 · · · tn = bt

Differencebetweenestimates

✻

❄

|f(b) − f(a)|

✲✛∆t

✲✛∆t

✻

❄

f(a)

✻

❄

f(b)

Figure 5.10: Left and right sums if f is decreasing

Accuracy of Estimates

For either increasing or decreasing velocity functions, the exact value of the distance traveled liessomewhere between the two estimates. Thus, the accuracy of our estimate depends on how closethese two sums are. For a function which is increasing throughout or decreasing throughout theinterval [a, b]:

!!!!!Difference between

upper and lower estimates

!!!!! =


f(a) and f(b)

!!!!! · ∆t = |f(b) − f(a)| · ∆t.

(Absolute values make the differences nonnegative.) In Figure 5.10, the area of the light rectanglesis the difference between estimates. By making the time interval, ∆t, between measurements smallenough, we can make this difference between lower and upper estimates as small as we like.

Exercises and Problems for Section 5.1Exercises

1. Figure 5.11 shows the velocity of a car for 0 ≤ t ≤ 12and the rectangles used to estimate of the distance trav-

eled.


(b) Do the rectangles lead to an upper or a lower esti-

mate?


(d) What is the value of ∆t?(e) Give an approximate value for the estimate.

120

4

t

Figure 5.112. The velocity v(t) in Table 5.3 is increasing, 0 ≤ t ≤ 12.

(a) Find an upper estimate for the total distance traveled

using

(i) n = 4 (ii) n = 2

(b) Which of the two answers in part (a) is more accu-

rate? Why?

(c) Find a lower estimate of the total distance traveled

using n = 4.

Table 5.3

t 0 3 6 9 12

v(t) 34 37 38 40 45

3. The velocity v(t) in Table 5.4 is decreasing, 2 ≤ t ≤ 12.

Using n = 5 subdivisions to approximate the total dis-

tance traveled, find

(a) An upper estimate (b) A lower estimate

Table 5.4

t 2 4 6 8 10 12

v(t) 44 42 41 40 37 35

2. The velocity v(t) in the table below is decreasing, 2 ≤ t ≤ 12. Using n = 5 subdivisions toapproximate the total distance traveled, find

(a) An upper estimate.

(b) A lower estimate.

5.1 HOW DO WE MEASURE DISTANCE TRAVELED? 277

v = f(t)

v

a = t0 t1 t2 · · · tn = bt

Differencebetweenestimates

✻

❄

|f(b) − f(a)|

✲✛∆t

✲✛∆t

✻

❄

f(a)

✻

❄

f(b)

Figure 5.10: Left and right sums if f is decreasing

Accuracy of Estimates

For either increasing or decreasing velocity functions, the exact value of the distance traveled liessomewhere between the two estimates. Thus, the accuracy of our estimate depends on how closethese two sums are. For a function which is increasing throughout or decreasing throughout theinterval [a, b]:


upper and lower estimates

!!!!! =


f(a) and f(b)

!!!!! · ∆t = |f(b) − f(a)| · ∆t.

(Absolute values make the differences nonnegative.) In Figure 5.10, the area of the light rectanglesis the difference between estimates. By making the time interval, ∆t, between measurements smallenough, we can make this difference between lower and upper estimates as small as we like.


1. Figure 5.11 shows the velocity of a car for 0 ≤ t ≤ 12and the rectangles used to estimate of the distance trav-

eled.


(b) Do the rectangles lead to an upper or a lower esti-

mate?


(d) What is the value of ∆t?(e) Give an approximate value for the estimate.

120

4

t

Figure 5.112. The velocity v(t) in Table 5.3 is increasing, 0 ≤ t ≤ 12.

(a) Find an upper estimate for the total distance traveled

using

(i) n = 4 (ii) n = 2

(b) Which of the two answers in part (a) is more accu-

rate? Why?

(c) Find a lower estimate of the total distance traveled

using n = 4.

Table 5.3

t 0 3 6 9 12

v(t) 34 37 38 40 45

3. The velocity v(t) in Table 5.4 is decreasing, 2 ≤ t ≤ 12.

Using n = 5 subdivisions to approximate the total dis-

tance traveled, find

(a) An upper estimate (b) A lower estimate

Table 5.4

t 2 4 6 8 10 12

v(t) 44 42 41 40 37 35

3. The figures below show the velocity, in cm/sec, of a particle moving along a number line. (Positivevelocities represent movement to the right; negative velocities to the left.) Compute the change inposition between times t = 0 and t = 5 seconds.

278 Chapter Five KEY CONCEPT: THE DEFINITE INTEGRAL

4. A car comes to a stop five seconds after the driver applies

the brakes. While the brakes are on, the velocities in the

table are recorded.

(a) Give lower and upper estimates of the distance the

car traveled after the brakes were applied.

(b) On a sketch of velocity against time, show the lower

and upper estimates of part (a).

(c) Find the difference between the estimates. Explain

how this difference can be visualized on the graph in

part (b).

Time since brakes applied (sec) 0 1 2 3 4 5

Velocity (ft/sec) 88 60 40 25 10 0

5. Figure 5.12 shows the velocity, v, of an object (in me-

ters/sec). Estimate the total distance the object traveled

between t = 0 and t = 6.

1 2 3 4 5 6

10

20

30

40

t (sec)

v (m/sec)

Figure 5.12

6. At time, t, in seconds, your velocity, v, in meters/second,

is given by

v(t) = 1 + t2 for 0 ≤ t ≤ 6.

Use ∆t = 2 to estimate the distance traveled during this

time. Find the upper and lower estimates, and then aver-

age the two.

7. Figure 5.13 shows the velocity of a particle, in cm/sec,

along a number line for time −3 ≤ t ≤ 3.

(a) Describe the motion in words: Is the particle chang-

ing direction or always moving in the same direc-

tion? Is the particle speeding up or slowing down?

(b) Make over and underestimates of the distance trav-

eled for −3 ≤ t ≤ 3.

−4 −3 −2 −1 0 1 2 3 4

1

2

4

t

Figure 5.13

8. For time, t, in hours, 0 ≤ t ≤ 1, a bug is crawling at a

velocity, v, in meters/hour given by

v =1

1 + t.

Use ∆t = 0.2 to estimate the distance that the bug crawls

during this hour. Find an overestimate and an underesti-

mate. Then average the two to get a new estimate.

Exercises 9–12 show the velocity, in cm/sec, of a particle mov-

ing along a number line. (Positive velocities represent move-

ment to the right; negative velocities to the left.) Compute the

change in position between times t = 0 and t = 5 seconds.

9.

3 5

2

−3

v(t)

t

10.

5

10

v(t)

t

11.

5

−3v(t)

t

12.

4

5

−2

8

v(t)

t

13. Use the expressions for left and right sums on page 276

and Table 5.5.

(a) If n = 4, what is ∆t? What are t0, t1, t2, t3, t4?

What are f(t0), f(t1), f(t2), f(t3), f(t4)?

(b) Find the left and right sums using n = 4.

(c) If n = 2, what is ∆t? What are t0, t1, t2? What are

f(t0), f(t1), f(t2)?

(d) Find the left and right sums using n = 2.

Table 5.5

t 15 17 19 21 23

f(t) 10 13 18 20 30


and Table 5.6.





f(t0), f(t1), f(t2)?


Table 5.6

t 0 4 8 12 16

f(t) 25 23 22 20 17




table are recorded.







part (b).


Velocity (ft/sec) 88 60 40 25 10 0




1 2 3 4 5 6

10

20

30

40

t (sec)

v (m/sec)

Figure 5.12


is given by

v(t) = 1 + t2 for 0 ≤ t ≤ 6.



age the two.








−4 −3 −2 −1 0 1 2 3 4

1

2

4

t

Figure 5.13



v =1

1 + t.








9.

3 5

2

−3

v(t)

t

10.

5

10

v(t)

t

11.

5

−3v(t)

t

12.

4

5

−2

8

v(t)

t


and Table 5.5.





f(t0), f(t1), f(t2)?


Table 5.5

t 15 17 19 21 23

f(t) 10 13 18 20 30


and Table 5.6.





f(t0), f(t1), f(t2)?


Table 5.6

t 0 4 8 12 16

f(t) 25 23 22 20 17

4. Find the difference between the upper and lower estimates of the distance traveled at velocityf(t) = 5t + 8 on the interval 1 ≤ t ≤ 3 for 100 subdivisions.

Problems are from Calculus by Hughes-Hallett, Gleason, et al., 7th Edition.


5. A woman drives 10 miles, accelerating uniformly from rest to 60 mph. Graph her velocity versustime. How long does it take for her to reach 60 mph?

6. Use the expressions for left and right sums and the table below

(a) If n = 4, what is ∆t? What are t0, t1, t2, t3, t4? What are f(t0), f(t1), f(t2), f(t3), f(t4)?


(c) If n = 2, what is ∆t? What are t0, t1, t2? What are f(t0), f(t1), f(t2)?





table are recorded.







part (b).


Velocity (ft/sec) 88 60 40 25 10 0




1 2 3 4 5 6

10

20

30

40

t (sec)

v (m/sec)

Figure 5.12


is given by

v(t) = 1 + t2 for 0 ≤ t ≤ 6.



age the two.








−4 −3 −2 −1 0 1 2 3 4

1

2

4

t

Figure 5.13



v =1

1 + t.








9.

3 5

2

−3

v(t)

t

10.

5

10

v(t)

t

11.

5

−3v(t)

t

12.

4

5

−2

8

v(t)

t


and Table 5.5.





f(t0), f(t1), f(t2)?


Table 5.5

t 15 17 19 21 23

f(t) 10 13 18 20 30


and Table 5.6.





f(t0), f(t1), f(t2)?


Table 5.6

t 0 4 8 12 16

f(t) 25 23 22 20 17

7. This problem concerns hybrid cars such as the Toyota Prius that are powered by a gas-engine,electric-motor combination, but can also function in Electric-Vehicle (EV) only mode. The figurebelow shows the velocity, v, of a 2010 Prius Plug-in Hybrid Prototype operating in normal hybridmode and EV-only mode, respectively, while accelerating from a stoplight.


26. Two cars start at the same time and travel in the same

direction along a straight road. Figure 5.17 gives the ve-

locity, v, of each car as a function of time, t. Which car:

(a) Attains the larger maximum velocity?

(b) Stops first?

(c) Travels farther?

Car A

Car B

t (hr)

v (km/hr)

Figure 5.17

27. A car initially going 50 ft/sec brakes at a constant rate

(constant negative acceleration), coming to a stop in 5seconds.

(a) Graph the velocity from t = 0 to t = 5.

(b) How far does the car travel?

(c) How far does the car travel if its initial velocity is

doubled, but it brakes at the same constant rate?

28. A woman drives 10 miles, accelerating uniformly from

rest to 60 mph. Graph her velocity versus time. How long

does it take for her to reach 60 mph?

29. An object has zero initial velocity and a constant accel-

eration of 32 ft/sec2. Find a formula for its velocity as a

function of time. Use left and right sums with ∆t = 1to find upper and lower bounds on the distance that the

object travels in four seconds. Find the precise distance

using the area under the curve.

Problems 30–31 concern hybrid cars such as the Toyota Prius

that are powered by a gas-engine, electric-motor combination,

but can also function in Electric-Vehicle (EV) only mode. Fig-

ure 5.18 shows the velocity, v, of a 2010 Prius Plug-in Hy-

brid Prototype operating in normal hybrid mode and EV-only

mode, respectively, while accelerating from a stoplight.1

5 15 25

40

80 Normal hybrid

EV-only

t (sec)

v (mph)

Figure 5.18

30. Could the car travel half a mile in EV-only mode during

the first 25 seconds of movement?

31. Assume two identical cars, one running in normal hybrid

mode and one running in EV-only mode, accelerate to-

gether in a straight path from a stoplight. Approximately

how far apart are the cars after 15 seconds?

Strengthen Your Understanding

In Problems 32–33, explain what is wrong with the statement.

32. If a car accelerates from 0 to 50 ft/sec in 10 seconds, then

it travels 250 ft.

33. For any acceleration, you can estimate the total distance

traveled by a car in 1 second to within 0.1 feet by record-

ing its velocity every 0.1 second.

In Problems 34–35, give an example of:

34. A velocity function f and an interval [a, b] such that the

distance denoted by the right-hand sum for f on [a, b] is

less than the distance denoted by the left-hand sum, no

matter what the number of subdivisions.

35. A velocity f(t) and an interval [a, b] such that at least

100 subdivisions are needed in order for the difference

between the upper and lower estimates to be less than or

equal to 0.1.

Are the statements in Problems 36–38 true or false? Give an

explanation for your answer.

36. For an increasing velocity function on a fixed time in-

terval, the left-hand sum with a given number of subdi-

visions is always less than the corresponding right-hand

sum.

37. For a decreasing velocity function on a fixed time inter-

val, the difference between the left-hand sum and right-

hand sum is halved when the number of subdivisions is

doubled.

38. For a given velocity function on a given interval, the dif-

ference between the left-hand sum and right-hand sum

gets smaller as the number of subdivisions gets larger.

39. A bicyclist starts from home and rides back and forth

along a straight east/west highway. Her velocity is given

1www.motortrend.com/, accessed May 2011.

(a) Could the car travel half a mile in EV-only mode during the first 25 seconds of movement?

(b) Assume two identical cars, one running in normal hybrid mode and one running in EV-onlymode, accelerate together in a straight path from a stoplight. Approximately how far apartare the cars after 15 seconds?

5.2: The Definite Integral

1. Use the table to estimate∫ 40

0f(x) dx. What values of n and ∆x did you use?



In Exercises 1–2, rectangles have been drawn to approximate! 6

0g(x) dx.




(d) What is the value of ∆x?

1.

6

g(x)

x

2.

6

g(x)

x

3. Figure 5.29 shows a Riemann sum approximation with n

subdivisions to! b

af(x) dx.

(a) Is it a left- or right-hand approximation? Would the

other one be larger or smaller?

(b) What are a, b, n and ∆x?

20x

Figure 5.29

4. Using Figure 5.30, draw rectangles representing each of

the following Riemann sums for the function f on the

interval 0 ≤ t ≤ 8. Calculate the value of each sum.

(a) Left-hand sum with ∆t = 4(b) Right-hand sum with ∆t = 4(c) Left-hand sum with ∆t = 2(d) Right-hand sum with ∆t = 2

2 4 6 8

48

121620242832

f(t)

t

Figure 5.30

In Exercises 5–10, use a calculator or a computer to find the

value of the definite integral.

5.

" 4

1

(x2 + x) dx 6.

" 3

0

2xdx

7.

" 1

−1

e−x2

dx 8.

" 3

0

ln(y2 + 1) dy

9.

" 1

0

sin(t2)dt 10.

" 4

3

√ez + z dz

11. Use the table to estimate! 40

0f(x)dx. What values of n

and ∆x did you use?

x 0 10 20 30 40

f(x) 350 410 435 450 460


0f(x) dx.

x 0 3 6 9 12

f(x) 32 22 15 11 9


0f(x) dx.

x 0 3 6 9 12 15

f(x) 50 48 44 36 24 8

14. Write out the terms of the right-hand sum with n = 5

that could be used to approximate

" 7

3

1

1 + xdx. Do not

evaluate the terms or the sum.

15. Use Figure 5.31 to estimate! 20

0f(x) dx.

4 8 12 16 20

1

2

3

4

5

f(x)

x

Figure 5.31

2. Use the figure to estimate∫ 20

0f(x) dx.





In Exercises 1–2, rectangles have been drawn to approximate! 6

0g(x) dx.




(d) What is the value of ∆x?

1.

6

g(x)

x

2.

6

g(x)

x

3. Figure 5.29 shows a Riemann sum approximation with n

subdivisions to! b

af(x) dx.

(a) Is it a left- or right-hand approximation? Would the

other one be larger or smaller?

(b) What are a, b, n and ∆x?

20x

Figure 5.29

4. Using Figure 5.30, draw rectangles representing each of

the following Riemann sums for the function f on the

interval 0 ≤ t ≤ 8. Calculate the value of each sum.

(a) Left-hand sum with ∆t = 4(b) Right-hand sum with ∆t = 4(c) Left-hand sum with ∆t = 2(d) Right-hand sum with ∆t = 2

2 4 6 8

48

121620242832

f(t)

t

Figure 5.30

In Exercises 5–10, use a calculator or a computer to find the

value of the definite integral.

5.

" 4

1

(x2 + x) dx 6.

" 3

0

2xdx

7.

" 1

−1

e−x2

dx 8.

" 3

0

ln(y2 + 1) dy

9.

" 1

0

sin(t2)dt 10.

" 4

3

√ez + z dz


0f(x)dx. What values of n

and ∆x did you use?

x 0 10 20 30 40

f(x) 350 410 435 450 460


0f(x) dx.

x 0 3 6 9 12

f(x) 32 22 15 11 9


0f(x) dx.

x 0 3 6 9 12 15

f(x) 50 48 44 36 24 8

14. Write out the terms of the right-hand sum with n = 5

that could be used to approximate

" 7

3

1

1 + xdx. Do not

evaluate the terms or the sum.


0f(x) dx.

4 8 12 16 20

1

2

3

4

5

f(x)

x

Figure 5.313. (a) What is the area between the graph of f(x) and the x−axis, between x = 0 and x = 5?

(b) What is∫ 5

0f(x) dx?

5.2 THE DEFINITE INTEGRAL 287


−10f(x)dx.

−10 0 10

10

20

30

x

f(x)

Figure 5.32

17. Using Figure 5.33, estimate! 5

−3f(x)dx.

−3 −1 2 4

5

−20

−10

10

x

f(x)

Figure 5.33

Problems

18. The graph of f(t) is in Figure 5.34. Which of the fol-

lowing four numbers could be an estimate of! 1

0f(t)dt

accurate to two decimal places? Explain your choice.

I. −98.35 II. 71.84 III. 100.12 IV. 93.47

0.5 1.0

20

40

60

80

100 f(t)

t

Figure 5.34

19. (a) What is the area between the graph of f(x) in Fig-

ure 5.35 and the x-axis, between x = 0 and x = 5?

(b) What is! 5

0f(x) dx?

3

5

✠

Area = 7

✒

Area = 6

x

f(x)

Figure 5.35

20. Find the total area between y = 4 − x2 and the x-axis

for 0 ≤ x ≤ 3.

21. (a) Find the total area between f(x) = x3 − x and the

x-axis for 0 ≤ x ≤ 3.

(b) Find

" 3

0

f(x)dx.

(c) Are the answers to parts (a) and (b) the same? Ex-

plain.

In Problems 22–28, find the area of the regions between the

curve and the horizontal axis

22. Under y = 6x3 − 2 for 5 ≤ x ≤ 10.

23. Under the curve y = cos t for 0 ≤ t ≤ π/2.

24. Under y = ln x for 1 ≤ x ≤ 4.

25. Under y = 2 cos(t/10) for 1 ≤ t ≤ 2.

26. Under the curve y = cos√

x for 0 ≤ x ≤ 2.

27. Under the curve y = 7 − x2 and above the x-axis.

28. Above the curve y = x4 − 8 and below the x-axis.

29. Use Figure 5.36 to find the values of

(a)! b

af(x) dx (b)

! c

bf(x) dx

(c)! c

af(x) dx (d)

! c

a|f(x)| dx

a b c

f(x)

✠

Area = 13

■Area = 2

x

Figure 5.36

30. Given! 0

−2f(x)dx = 4 and Figure 5.37, estimate:

(a)! 2

0f(x)dx (b)

! 2

−2f(x)dx

(c) The total shaded area.

−2 2−2

2

f(x)

x

Figure 5.37

4. (a) Use the figure below to find∫ 0

−3 f(x) dx.?

(b) If the area of the shaded region is A, estimate∫ 4

−3 f(x) dx?


31. (a) Using Figure 5.38, find! 0

−3f(x) dx.

(b) If the area of the shaded region is A, estimate! 4

−3f(x) dx.

−4 −3 −2 −1 1 2 3

4

5

−1

1

x

f(x)

Figure 5.38

32. Use Figure 5.39 to find the values of

(a)! 2

0f(x) dx (b)

! 7

3f(x) dx

(c)! 7

2f(x) dx (d)

! 8

5f(x) dx

2 4 6 8 10−2

−1

1

2f(x)

x

Figure 5.39: Graph consists of a semicircle and

line segments

33. (a) Graph f(x) = x(x + 2)(x − 1).

(b) Find the total area between the graph and the x-axis

between x = −2 and x = 1.

(c) Find! 1

−2f(x) dx and interpret it in terms of areas.

34. Compute the definite integral! 4

0cos

√x dx and interpret

the result in terms of areas.

35. Without computation, decide if! 2π

0e−x sin x dx is posi-

tive or negative. [Hint: Sketch e−x sin x.]

36. Estimate! 1

0e−x2

dx using n = 5 rectangles to form a

(a) Left-hand sum (b) Right-hand sum

37. (a) On a sketch of y = lnx, represent the left Riemann

sum with n = 2 approximating! 2

1ln x dx. Write

out the terms in the sum, but do not evaluate it.

(b) On another sketch, represent the right Riemann sum

with n = 2 approximating! 2

1ln x dx. Write out the

terms in the sum, but do not evaluate it.

(c) Which sum is an overestimate? Which sum is an un-

derestimate?

38. (a) Draw the rectangles that give the left-hand sum ap-

proximation to! π

0sin x dx with n = 2.

(b) Repeat part (a) for! 0

−πsin x dx.

(c) From your answers to parts (a) and (b), what is

the value of the left-hand sum approximation to! π

−πsin x dx with n = 4?

39. (a) Use a calculator or computer to find! 6

0(x2 + 1) dx.

Represent this value as the area under a curve.

(b) Estimate! 6

0(x2 + 1) dx using a left-hand sum with

n = 3. Represent this sum graphically on a sketch

of f(x) = x2 + 1. Is this sum an overestimate or

underestimate of the true value found in part (a)?

(c) Estimate! 6

0(x2+1) dx using a right-hand sum with

n = 3. Represent this sum on your sketch. Is this

sum an overestimate or underestimate?

40. (a) Graph f(x) ="

1 − x 0 ≤ x ≤ 1x − 1 1 < x ≤ 2.

(b) Find

# 2

0

f(x) dx.

(c) Calculate the 4-term left Riemann sum approxima-

tion to the definite integral. How does the approxi-

mation compare to the exact value?

41. Estimate! 2

1x2 dx using left- and right-hand sums with

four subdivisions. How far from the true value of the in-

tegral could your estimate be?

42. Without computing the sums, find the difference between

the right- and left-hand Riemann sums if we use n = 500

subintervals to approximate! 1

−1(2x3 + 4) dx.

43. Sketch the graph of a function f (you do not need to give

a formula for f ) on an interval [a, b] with the property

that with n = 2 subdivisions,

# b

a

f(x) dx < Left-hand sum < Right-hand sum.

44. Write a few sentences in support of or in opposition to

the following statement:

“If a left-hand sum underestimates a definite integral

by a certain amount, then the corresponding right-hand

sum will overestimate the integral by the same amount.”

45. Consider the integral! 2

1(1/t) dt in Example 1. By divid-

ing the interval 1 ≤ t ≤ 2 into 10 equal parts, we can

show that

0.1$

1

1.1+

1

1.2+ . . . +

1

2

%≤

# 2

1

1

tdt

and# 2

1

1

tdt ≤ 0.1

$1

1+

1

1.1+ . . . +

1

1.9

%.

(a) Now divide the interval 1 ≤ t ≤ 2 into n equal parts

to show that

n&

r=1

1

n + r<

# 2

1

1

tdt <

n−1&

r=0

1

n + r.

(b) Show that the difference between the upper and

lower sums in part (a) is 1/(2n).

(c) The exact value of! 2

1(1/t) dt is ln 2. How large

should n be to approximate ln 2 with an error of at

most 5 · 10−6, using one of the sums in part (a)?

5. Without computation, decide of∫ 2π

0e−x sin(x) dx is positive or negative.

6. Sketch the graph of a function f (you do not need to give a formula for f) on an interval [a, b] with

the property that with n = 2 subdivisions,∫ baf(x) dx < Left-hand sum < Right-hand sum.

7. Explain what is wrong with the statement:

(a) For any function,∫ 3

1f(x) dx is the area between the graph of f and the x−axis on 1 ≤ x ≤ 3.

(b) The left hand sum with 10 subdivisions for the integral∫ 2

1sin(x) dx is

0.1(

sin(1.1) + ... + sin(2)).



8. Give an example of:

(a) A function f and an interval [a, b] such that∫ baf(x) dx is negative.

(b) A function f such that∫ 3

1f(x) dx <

∫ 2

1f(x) dx.

5.3: The Fundamental Theorem of Calculus

1. If f(t) is measured in dollars per year and t is measured in years, what are the units of∫ baf(t) dt?

2. If f(x) is measured in pounds and x is measured in feet, what are the units of∫ baf(x) dx?

3. Explain in words what the integral represents and give units:∫ 6

0a(t) dt, where a(t) is acceleration

in km/hr2 and t is time in hours.

4. Let f(t) = F ′(t). Write the integral∫ baf(t) dt and evaluate it using the Fundamental Theorem of

Calculus for F (t) = 7 · 4t; a = 2, b = 3.

5. Pollution is removed from a lake on day t at a rate of f(t) kg/day.

(a) Explain the meaning of the statement f(12) = 500.

(b) If∫ 15

5f(t) dt = 4000, give units of the 5, the 15, and the 4000.

(c) Give the meaning of∫ 15

5f(t) dt = 4000.

6. Water is leaking out of a tank at a rate of R(t) gallons/hour, where t is measured in hours.

(a) Write a definite integral that expresses the total amount of water that leaks out in the firsttwo hours.

(b) In the figure below, shade the region whose area represents the total amount of water thatleaks out in the first two hours.

(c) Give an upper and lower estimate of the total amount of water that leaks out in the first twohours.

5.3 THE FUNDAMENTAL THEOREM AND INTERPRETATIONS 295

17. (a) If F (t) = 12

sin2 t, find F ′(t).

(b) Find

! 0.4

0.2

sin t cos t dt two ways:

(i) Numerically.

(ii) Using the Fundamental Theorem of Calculus.

18. (a) If F (x) = ex2

, find F ′(x).

(b) Find

! 1

0

2xex2

dx two ways:

(i) Numerically.

(ii) Using the Fundamental Theorem of Calculus.

19. Pollution is removed from a lake on day t at a rate of

f(t) kg/day.

(a) Explain the meaning of the statement f(12) = 500.

(b) If" 15

5f(t) dt = 4000, give the units of the 5, the

15, and the 4000.

(c) Give the meaning of" 15

5f(t) dt = 4000.

20. Oil leaks out of a tanker at a rate of r = f(t) gallons per

minute, where t is in minutes. Write a definite integral

expressing the total quantity of oil which leaks out of the

tanker in the first hour.

21. Water is leaking out of a tank at a rate of R(t) gal-

lons/hour, where t is measured in hours.

(a) Write a definite integral that expresses the total

amount of water that leaks out in the first two hours.

(b) In Figure 5.42, shade the region whose area repre-

sents the total amount of water that leaks out in the

first two hours.

(c) Give an upper and lower estimate of the total amount

of water that leaks out in the first two hours.

1 2

2

1

t

R(t)

Figure 5.42

22. As coal deposits are depleted, it becomes necessary to

strip-mine larger areas for each ton of coal. Figure 5.43

shows the number of acres of land per million tons of coal

that will be defaced during strip-mining as a function of

the number of million tons removed, starting from the

present day.

(a) Estimate the total number of acres defaced in ex-

tracting the next 4 million tons of coal (measured

from the present day). Draw four rectangles under

the curve, and compute their area.

(b) Re-estimate the number of acres defaced using rect-

angles above the curve.

(c) Use your answers to parts (a) and (b) to get a better

estimate of the actual number of acres defaced.

1 2 3 4 5

1

2

3

4

0.2

million tons ofcoal extracted(measuredfrom presentday)

acres defacedper million tons

Figure 5.43

23. The rate at which the world’s oil is consumed (in billions

of barrels per year) is given by r = f(t), where t is in

years and t = 0 is the start of 2004.

(a) Write a definite integral representing the total quan-

tity of oil consumed between the start of 2004 and

the start of 2009.

(b) Between 2004 and 2009, the rate was modeled by

r = 32e0.05t. Using a left-hand sum with five subdi-

visions, find an approximate value for the total quan-

tity of oil consumed between the start of 2004 and

the start of 2009.

(c) Interpret each of the five terms in the sum from

part (b) in terms of oil consumption.

24. A bungee jumper leaps off the starting platform at time

t = 0 and rebounds once during the first 5 seconds.

With velocity measured downward, for t in seconds and

0 ≤ t ≤ 5, the jumper’s velocity is approximated5 by

v(t) = −4t2 + 16t meters/sec.

(a) How many meters does the jumper travel during the

first five seconds?

(b) Where is the jumper relative to the starting position

at the end of the five seconds?

(c) What does" 5

0v(t) dt represent in terms of the

jump?

25. An old rowboat has sprung a leak. Water is flowing into

the boat at a rate, r(t), given in the table.

(a) Compute upper and lower estimates for the volume

of water that has flowed into the boat during the 15minutes.

(b) Draw a graph to illustrate the lower estimate.

t minutes 0 5 10 15

r(t) liters/min 12 20 24 16

5Based on www.itforus.oeiizk.waw.pl/tresc/activ//modules/bj.pdf. Accessed Feb 12, 2012.



7. The graph of a continuous function f is given in the figure below. Rank the following integrals inascending numerical order. Explain your reasons.

(a)∫ 2

0f(x) dx

(b)∫ 1

0f(x) dx

(c)∫ 2

0

(f(x)

)1/2dx

(d)∫ 2

0

(f(x)

)2dx


26. Annual coal production in the US (in billion tons per

year) is given in the table.6 Estimate the total amount

of coal produced in the US between 1997 and 2009. If

r = f(t) is the rate of coal production t years since

1997, write an integral to represent the 1997–2009 coal

production.

Year 1997 1999 2001 2003 2005 2007 2009

Rate 1.090 1.094 1.121 1.072 1.132 1.147 1.073

27. The amount of waste a company produces, W , in tons per

week, is approximated by W = 3.75e−0.008t , where t is

in weeks since January 1, 2005. Waste removal for the

company costs $15/ton. How much does the company

pay for waste removal during the year 2005?

28. A two-day environmental cleanup started at 9 am on the

first day. The number of workers fluctuated as shown in

Figure 5.44. If the workers were paid $10 per hour, how

much was the total personnel cost of the cleanup?

8 16 24 32 40 48

10

20

30

40

50

hours

workers

Figure 5.44

29. Suppose in Problem 28 that the workers were paid $10

per hour for work during the time period 9 am to 5 pm

and were paid $15 per hour for work during the rest of the

day. What would the total personnel costs of the cleanup

have been under these conditions?

30. A warehouse charges its customers $5 per day for ev-

ery 10 cubic feet of space used for storage. Figure 5.45

records the storage used by one company over a month.

How much will the company have to pay?

10 20 30

10,000

20,000

30,000

days

cubic feet

Figure 5.45

31. A cup of coffee at 90◦C is put into a 20◦C room when

t = 0. The coffee’s temperature is changing at a rate of

r(t) = −7e−0.1t ◦C per minute, with t in minutes. Esti-

mate the coffee’s temperature when t = 10.

32. Water is pumped out of a holding tank at a rate of

5 − 5e−0.12t liters/minute, where t is in minutes since

the pump is started. If the holding tank contains 1000liters of water when the pump is started, how much water

does it hold one hour later?

Problems 33–34 concern the graph of f ′ in Figure 5.46.

1 2 3 4x

f ′(x)

Figure 5.46: Graph of f ′, not f

33. Which is greater, f(0) or f(1)?

34. List the following in increasing order:f(4) − f(2)

2, f(3) − f(2), f(4) − f(3).

35. A force F parallel to the x-axis is given by the graph in

Figure 5.47. Estimate the work, W , done by the force,

where W =! 16

0F (x) dx.

4 8 10

14 16

−2

−1

1

2

x (meter)

force (newton)

F

Figure 5.47

36. Let f(1) = 7, f ′(t) = e−t2 . Use left- and right-hand

sums of 5 rectangles each to estimate f(2).

37. The graph of a continuous function f is given in Fig-

ure 5.48. Rank the following integrals in ascending nu-

merical order. Explain your reasons.

(i)! 2

0f(x) dx (ii)

! 1

0f(x) dx

(iii)! 2

0(f(x))1/2 dx (iv)

! 2

0(f(x))2 dx.

0 1 2

100

x

f(x)

Figure 5.48

6http://www.eia.doe.gov/cneaf/coal/page/special/tbl1.html. Accessed May 2011.

8. The graphs in the figure below represent the velocity, v, of a particle moving along the x-axis fortime 0 ≤ t ≤ 5. The vertical scales of all graphs are the same. Identify the graph showing whichparticle:

(a) Has a constant acceleration.

(b) Ends up farthest to the left of where it started.

(c) Ends up the farthest from its starting point.

(d) Experiences the greatest initial acceleration.

(e) Has the greatest average velocity.

(f) Has the greatest average acceleration.

5.3 THE FUNDAMENTAL THEOREM AND INTERPRETATIONS 297

38. The graphs in Figure 5.49 represent the velocity, v, of a

particle moving along the x-axis for time 0 ≤ t ≤ 5.

The vertical scales of all graphs are the same. Identify

the graph showing which particle:

(a) Has a constant acceleration.

(b) Ends up farthest to the left of where it started.

(c) Ends up the farthest from its starting point.

(d) Experiences the greatest initial acceleration.

(e) Has the greatest average velocity.

(f) Has the greatest average acceleration.

5t

v(I)

t

v

5

(II)

t

v

5

(III)

t

v

5

(IV)

t

v

5

(V)

Figure 5.49

39. A car speeds up at a constant rate from 10 to 70 mph

over a period of half an hour. Its fuel efficiency (in miles

per gallon) increases with speed; values are in the table.

Make lower and upper estimates of the quantity of fuel

used during the half hour.

Speed (mph) 10 20 30 40 50 60 70

Fuel efficiency (mpg) 15 18 21 23 24 25 26

In Problems 40–41, oil is pumped from a well at a rate of

r(t) barrels per day. Assume that t is in days, r′(t) < 0 and

t0 > 0.

40. What does the value of! t0

0r(t) dt tells us about the oil

well?

41. Rank in order from least to greatest:

" 2t0

0

r(t) dt,

" 2t0

t0

r(t) dt,

" 3t0

2t0

r(t) dt.

42. Height velocity graphs are used by endocrinologists to

follow the progress of children with growth deficiencies.

Figure 5.50 shows the height velocity curves of an aver-

age boy and an average girl between ages 3 and 18.

(a) Which curve is for girls and which is for boys? Ex-

plain how you can tell.

(b) About how much does the average boy grow be-

tween ages 3 and 10?

(c) The growth spurt associated with adolescence and

the onset of puberty occurs between ages 12 and 15

for the average boy and between ages 10 and 12.5

for the average girl. Estimate the height gained by

each average child during this growth spurt.

(d) When fully grown, about how much taller is the av-

erage man than the average woman? (The average

boy and girl are about the same height at age 3.)

2 4 6 8 10 12 14 16 18

2

4

6

8

10

x (years)

y (cm/yr)

Figure 5.50

In Problems 43–45, evaluate the expressions using Table 5.8.

Give exact values if possible; otherwise, make the best possi-

ble estimates using left-hand Riemann sums.

Table 5.8

t 0.0 0.1 0.2 0.3 0.4 0.5

f(t) 0.3 0.2 0.2 0.3 0.4 0.5

g(t) 2.0 2.9 5.1 5.1 3.9 0.8

43.

" 0.5

0

f(t) dt 44.

" 0.5

0.2

g′(t) dt

45.

" 0.3

0

g (f(t)) dt


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