CALCULUS

WORKSHEET ON RIEMANN SUMS

Work the following on notebook paper. Use your calculator, and give decimal answers correct to three

decimal places.

On problems 1 – 2, estimate the area bounded by the curve and the x-axis on the given interval using the

indicated number of subintervals by finding:

(a) a left Riemann sum

(b) a right Riemann sum

(c) a midpoint Riemann sum

1. [0,1]y x n = 4 subintervals 2. 1

[1, 3]yx

n = 4 subintervals

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3. Estimate the area bounded by the curve and the x-axis on [1, 6] using the 5 equal subintervals by

finding:

(a) a left Riemann sum

(b) a right Riemann sum

(c) a midpoint Riemann sum

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4. Oil is leaking out of a tank. The rate of flow is measured every two hours for a 12-hour period,

and the data is listed in the table below.

Time (hr) 0 2 4 6 8 10 12

Rate (gal/hr) 40 38 36 30 26 18 8

(a) Draw a possible graph for the data given in the table.

(b) Estimate the number of gallons of oil that have leaked out of the tank during the 12-hour period

by finding a left Riemann sum with three equal subintervals.

(c) Estimate the number of gallons of oil that have leaked out of the tank during the 12-hour period

by finding a right Riemann sum with three equal subintervals.

(d) Estimate the number of gallons of oil that have leaked out of the tank during the 12-hour period

by finding a midpoint Riemann sum with three equal subintervals.

TURN->>>

5. Oil is being pumped into a tank over a 12-hour period. The tank contains 120 gallons of oil when t = 0.

The rate at which oil is flowing into the tank at various times is modeled by a differentiable function R

for 0 12,t where t is measured in hours and R t is measured in gallons per hours. Values of R t

at selected values of time t are shown in the table below.

t (hours) 0 3 5 9 12

R t (gallons per hour) 8.9 6.8 6.4 5.9 5.7

(a) Estimate the number of gallons of oil in the tank at t = 12 hours by using a left Riemann sum with four

subintervals and values from the table. Show the computations that lead to your answer.

(b) Estimate the number of gallons of oil in the tank at t = 12 hours by using a right Riemann sum with four

subintervals and values from the table. Show the computations that lead to your answer.

__________________________________________________________________________________________

6. A hot cup of coffee is taken into a classroom and set on a desk to cool. When t = 0, the temperature of the

coffee is 113° F. The rate at which the temperature of the coffee is dropping is modeled by a differentiable

function R for 0 8,t where R t is measured in degrees Fahrenheit per minute and t is measured in

minutes. Values of R t at selected values of time t are shown in the table below.

t (minutes) 0 3 5 8

R t (° F/min.) 5.5 2.7 1.6 0.8

(a) Estimate the temperature of the coffee at t = 8 minutes by using a left Riemann sum with three subintervals

and values from the table. Show the computations that lead to your answer.

(b) Estimate the temperature of the coffee at t = 8 minutes by using a right Riemann sum with three

subintervals and values from the table. Show the computations that lead to your answer.

__________________________________________________________________________________________

7. (Modification of 2004 Form B AB 3/ BC 3)

A test plane flies in a straight line with positive velocity v t , in miles per minute at time t minutes, where

v is a differentiable function of t. Selected values of v t for 0 40t are shown in the table below.

t (min) 0 5 10 15 20 25 30 35 40

v t (mpm) 7. 0 9.2 9.5 7. 0 4.5 2.4 2.4 4.3 7.2

Use a midpoint Riemann sum with four subintervals of equal length and values from the table to approximate

the distance traveled by the plane during the 40 minutes. Show the computations that lead to your answer.

Answers to Worksheet on Riemann Sums 1. (a) 0.518

(b) 0.768

(c) 0.673

2. (a) 1.283

(b) 0.95

(c) 1.090

3. (a) 14

(b) 13

(c) 13.5

4. (a) Graph

(b) 408 gal

(c) 280 gal

(d) 344 gal

5. (a) 203.6 gal

(b) 193.9 gal

6. (a) 86.3°F

(b) 99.3°F

7. 229 miles

AP CALCULUS

WORKSHEET ON DEFINITE INTEGRALS AND AREA

Work the following on notebook paper. Do not use your calculator except on problem 15.

Evaluate.

1. 31 2

01x x dx 5.

5

0

225 x dx 9. 2

0

2cos

3

xdx

2. 1 2 3

01x x dx 6.

4

0 29

xdx

x 10.

32

6

sin cosx x dx

3.

1

32 2 2

xdx

x

7. 6

0cos 3x dx

11. 24

0tan secx x dx

4. 4

12 1x dx

8. 6

12

sin 2x dx

12. 6

0sec 2 tan 2x x dx

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13. Find the area bounded by the graph of 2sin sin 2f x x x and the x-axis

on the interval 0, .

14. Find the area bounded by the graph of 2sec2

xf x

and the x-axis

on the interval 2

,2 3

.

________________________________________________________________________________________

Use your calculator on problem 15.

15. The rate at which water is being pumped into a tank is given by the function R t . A table of selected

values of R t , for the time interval 0 20t minutes, is shown below.

t (min.) 0 4 9 17 20

R t (gal/min) 25 28 33 42 46

(a) Use data from the table and four subintervals to find a left Riemann sum to approximate the value

of 20

0R t dt .

(b) Use data from the table and four subintervals to find a right Riemann sum to approximate the value

of 20

0R t dt .

(c) A model for the rate at which water is being pumped into the tank is given by the function

0.0325 tW t e , where t is measured in minutes and W t is measured in gallons per minute.

Use the model to find the value of 20

0W t dt .

Answers to Worksheet on Definite Integrals and Area

1. 15

8

2. 3

22

2 19

3. 1

48

4. 1

142

5. 25

4

6. 2

7. 1

3

8. 1 1 3

2 2 2

9. 3 3

4

10. 15

64

11. 2

3

12. 1

2

13. 4

14. 2 3 1

15. (a) 630 gal

(b) 751 gal

(c) 685.099 gal

CALCULUS

WORKSHEET ON ALGEBRAIC & U-SUBSTITUTION

Work the following on notebook paper. Do not use your calculator.

Evaluate.

1. 2x x dx 5. 2

2

4

xdx

x 9.

2 2

1

32 1x x dx

2. 2 2x x dx 6.

73

11x x dx

10.

4

0

1

2 1dx

x

3. 2 1x x dx 7.

31 2

11x x dx

11.

3

34 5

xdx

x

4. 4

xdx

x 8.

5

1 2 1

xdx

x

________________________________________________________________________________________

12. Find the area bounded by the graph of 3 1y x x and the x-axis on the

interval [0, 7].

13. Find the area bounded by the graph of cosy x x and the x-axis on the

interval ,3 2

.

________________________________________________________________________________________

14. Solve:

3

48 given that 1, 3

3 5

dy

dx x

is a point on the solution curve.

________________________________________________________________________________________

Given that f x is an even function and that 2

0

8

3f x dx , find:

15. 0

2f x dx

16. 2

2f x dx

17. 0

23 f x dx

________________________________________________________________________________________

Given that f x is an odd function and that 2

0

8

3f x dx , find:

18. 0

2f x dx

19. 2

2f x dx

20. 0

23 f x dx

________________________________________________________________________________________

21. Write

3 3 31 1 2 5

...limn

n

n n n n

as a definite integral, given that n is a positive integer.

22. The closed interval [c, d] is partitioned into n equal subintervals, each of width ,x by the numbers

1 1 2 10 0, , ..., where ...n nc x x x x x dx x . Write 1

2lim

n

i

kn

xx

as a definite integral.

x

y

x

y

x

y

CALCULUS

WORKSHEET 1 ON FUNDAMENTAL THEOREM OF CALCULUS

Work the following on notebook paper.

Work problems 1 - 2 by both methods. Do not use your calculator.

1. 2

12 and 1 6. Find 3 .y y y

x

2. cos 2 and 0 3. Find .4

f x x f f

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Work problems 3 – 7 using the Fundamental Theorem of Calculus and your calculator.

3. 3cos and 0 2. Find 1 .f x x f f

4. 2

and 5 1. Find 2 .xf x e f f

5. A particle moving along the x-axis has position x t at time t with the velocity of the particle

25sin .v t t At time t = 6, the particle’s position is (4, 0). Find the position of the particle

when t = 7.

6. Let F t represent a bacteria population which is 4 million at time t = 0. After t hours, the population is

growing at an instantaneous rate of 2t million bacteria per hour. Find the total increase in the bacteria

population during the first three hours, and find the population at t = 3 hours.

7. A particle moves along a line so that at any time 0t its velocity is given by 21

tv t

t

. At time

t = 0, the position of the particle is 0 5.s Determine the position of the particle at t = 3.

______________________________________________________________________________________

Use the Fundamental Theorem of Calculus and the given graph.

8. The graph of f is shown on the right.

4

16.2 and 1 3. Find 4 .f x dx f f

9. The graph of f is the semicircle shown on the right.

Find 4 given that 4 7.f f

10. The graph of f , consisting of two line segments

and a semicircle, is shown on the right. Given

that 2 5f , find:

(a) 1f (b) 4f (c) 8f

TURN->>>

4 1

4 - 4

11. Region A has an area of 1.5, and 6

03.5.f x dx Find:

(a) 6

2f x dx

(b) 6

0f x dx

12. The graph on the right shows the rate of

change of the quantity of water in a water

tower, in liters per day, during the month

of April. If the tower has 12,000 liters of

water in it on April 1, estimate the quantity

of water in the tower on April 30.

13. 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 0.37 tr t e C per minute, with t in minutes. Estimate the coffee’s temperature when t = 10.

14. Use the figure on the right and the

fact that 2 3F to sketch the

graph of .F x Label the values

of at least four points.

Answers to Worksheet 1 on the First Fundamental Theorem of Calculus

1. 32

3

2. 7

2

3. 2.932

4. 0.996

5. (3.837, 0)

6. 14.099 million, 10.099 million

7. 6.151

8. 9.2

9. 7 8

10. (a) 9.5 (b) 6.5 (c) 6.5 + 2π 11. (a) 5 (b) 6.5

12. Answers will vary. One approximation is 13,500 liters, found with two triangles and two trapezoids.

13. 67.828°C

14. (a) 0 1, 6 4, 8 0F F F

(b) F is increasing on 0 2 and 6 8x x b/c 0F x ther e.

F is decreasing on 2 6x b/c 0F x there.

(c) F is CU on 0 1 and 4.5 7x x b/c F x is increasing.

F is CD on 1 4.5 and 7 8x x b/c F x is decreasing.

CALCULUS

WORKSHEET 2 ON FUNDAMENTAL THEOREM OF CALCULUS

Work these on notebook paper. Use your calculator on problems 3, 8, and 13.

1. If 4

11 12, is continuous, and 17,f f f x dx what is the value of 4 ?f

2. If 5 5

2 22 3 17, find .f x dx f x dx

3. Water is pumped out of a holding tank at a rate of 0.125 5 te liters/minute, where t is in minutes since the

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

does it hold one hour later?

4. Given the values of the derivative f x in the table and that 0 100,f estimate f x for x = 2, 4, 6.

Use a right Riemann sum.

x 0 2 4 6

f x 10 18 23 25

5. Consider the function f that is continuous on the interval [ 5,5] and for which

5

04.f x dx Evaluate:

(a) 5

02f x dx (c)

5

5( is even) f x dx f

(b) 3

22f x dx

(d)

5

5( is odd) f x dx f

6. Use the figure on the right and the

fact that 0 2P to find values

of P when t = 1, 2, 3, 4, and 5.

7. In the figure on the right, the graph of g is given. Let be the antiderivative of .G t g t

(a) Given 0 5,G find 2 , 4 , and 5 .G G G

(b) Find the intervals where the graph of G is

increasing and decreasing. Justify your answer.

(c) Find the intervals where the graph of G is

concave up and concave down. Justify your answer.

(d) Sketch a graph of an antiderivative .G t Label

each critical point of G t with its coordinates.

TURN->>>

x

y

8. Find the value of 2

1 , where and 0 2.xF F x e F

5

12

2 , 19. Given . Evaluate: .

2, 1

x xf x f x dx

x

10. A bowl of soup is placed on the kitchen counter to cool. The temperature of the soup is given in the table

below.

Time t (minutes) 0 5 8 12

Temperature T x (°F) 105 99 97 93

(a) Find 12

0T x dx .

(b) Find the average rate of change of T x over the time interval t = 5 to t = 8 minutes.

11. The graph of f which consists of a line

segment and a semicircle, is shown on the

right. Given that 1 4,f find:

(a) 2f

(b) 5f

12. (Multiple Choice) If f and g are continuous functions such that g x f x for all x ,

then 3

2f x dx

(A) 2 3g g (B) 3 2g g (C) 3 2g g

(D) 3 2f f (E) 3 2f f

13. (Multiple Choice) If the function f x is defined by 3 2f x x and g is an antiderivatives

of f such that 3 5g , then 1g

(A) 3.268 (B) 1.585 (C) 1.732 (D) 6.585 (E) 11.585

14. (Multiple Choice) The graph of f is shown in the figure at right.

If 3

12.3f x dx and F x f x , then 3 0F F

(A) 0.3 (B) 1.3 (C) 3.3 (D) 4.3 (E) 5.3

Answers to Worksheet 2 on the First Fundamental Theorem

1. 29

2. 4 3.

741.636 liters

4. 136, 182, 232

5. (a) 14

(b) 4

(c) 8

(d) 0

6. 1, 0, 1

2 , 0, 1

7. (a) 2 21, 4 13, 5 15G G G

(b) G is increasing on 0 2 and 4 5x x b/c 0G x there.

G is decreasing on 2 4x b/c 0G x there.

(c) G is CU on 0 and 3 4.52

3x x b/c G x is increasing there.

G is CD on 3 and 4.5 52

3x x b/c G x is decreasing there.

(d) Graph

8. 2.747

9. 3

84

10. (a) 12°F (b) 2

3 °F per minute

11. (a) 7 (b) 12 2

12. C

13. B

14. D

CALCULUS

WORKSHEET ON AVERAGE VALUE

Work the following on notebook paper. Use your calculator on problems 3 – 6, and give decimal answers

correct to three decimal places.

On problems 1 and 2,

(a) Find the average value of f on the given interval.

(b) Find the value of c such that AVEf f c .

1. 2

3 , 2, 5f x x 2. , 0, 4f x x

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3. The table below gives values of a continuous function. Use a midpoint Riemann sum with three

equal subintervals to estimate the average value of f on [20, 50].

x 20 25 30 35 40 45 50

f x 42 38 31 29 35 48 60

__________________________________________________________________________________________

4. The velocity graph of an accelerating car is shown on the right.

(a) Estimate the average velocity of the car during the first

12 seconds by using a midpoint Riemann sum with three

equal subintervals.

(b) At what time was the instantaneous velocity equal to the

average velocity?

__________________________________________________________________________________________

5. In a certain city, the temperature, in °F, t hours after 9 AM was modeled by the function

50 14sin12

tT t

. Find the average temperature during the period from 9 AM to 9 PM.

__________________________________________________________________________________________

6. If a cup of coffee has temperature 95°C in a room where the temperature is 20°C, then, according

to Newton’s Law of Cooling, the temperature of the coffee after t minutes is given by the

function 5020 75t

T t e

. What is the average temperature of the coffee during the first half

hour?

__________________________________________________________________________________________

7. Suppose the C t represents the daily cost of heating your house, measured in dollars per day,

where t is time measured in days and t = 0 corresponds to January 1, 2010.. Interpret

90 90

0 0

1 and

90 0C t dt C t dt

.

TURN->>>

x

y

x

y

x

y

8. Using the figure on the right,

(a) Find 6

1f x dx .

(b) What is the average value of f on [1, 6]?

Graph of f

________________________________________________________________________________________

9. The average value of y f x equals 4 for 1 6x and equals 5 for 6 8x .

What is the average value of f x for 1 8x ?

________________________________________________________________________________________

In problems 10 – 11, find the average value of the function on the given interval without integrating.

Hint: Use Geometry. (No calculator)

10. 4, 4 1

on 4, 22, 1 2

x xf x

x x

11. 21 1 1, 1f x x

Answers to Worksheet on Average Value

1. 1. 2 and 4.avef c

2. 4 16

. .3 9

avef c

3. 1150 1

or 383 3

4. (a) km

45hr

(b) 5 sec

5. 58.913 F

6. 76.399 C

7. 90

0 represents the total cost in dollars to heat your house for the first 90 days of 2010.C t dt

90

0

1 represents the average cost in dollars per day to heat your house for the first 90 days of 2010

90 0C t dt

.

8. (a) 8.5 (b) 1.7

9. 30

7

10. 1.5

11. 14