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CALCULUS
WORKSHEET ON RIEMANN SUMS AND ANTIDERIVATIVES
Work the following on notebook paper. Use your calculator, and give decimal answers correct to three
decimal places.
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. y 
x
[0,1]
2. y 
n = 4 subintervals
1
x
[1, 3]
n = 4 subintervals
_______________________________________________________________________________________
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
_______________________________________________________________________________________
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)
Rate (gal/hr)
0
40
2
38
4
36
6
30
8
26
10
18
12
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.
_________________________________________________________________________________________
Evaluate.
5.
  x  1 2 x  3 dx
8.
3
 x  x  2  dx
11.
x
6.
x2  2 x  3
 x4 dx
9.
x
12.
3
 sin  5x  cos  5x  dx
7.
  5  sec x tan x  sec x  dx
2
10.
5
2

3
x 4  2 dx
x4
x2  8x  1
2


cos 2 x3 dx
dx
_________________________________________________________________________________________
13. Find a function f such that the graph of f has a horizontal tangent at (2, 0) and f   x   6 x .
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.
 
2.
0 x
3.
4.
1
0

5.
x3  1 dx
6.
3
x x2  1 dx
1 2
1
2
4
 1
dx
3

7.
2 x  1 dx
8.
2
2
0

9 x
6
9.
0
10.
 6
2
dx

cos  3x  dx
11.
sin  2x  dx
12.
  12
6
2

x
0
0

25  x 2 dx
4

x
x
5
0

0
2
 2x 
cos   dx
 3 
sin 3 x cos x dx
4
tan x sec 2 x dx
6
sec  2 x  tan  2 x  dx
________________________________________________________________________________________
13. Find the area bounded by the graph of f  x   2sin x  sin  2 x  and the x-axis
on the interval 0,   .
x
 and the x-axis
2
14. Find the area bounded by the graph of f  x   sec 2 
  2 
on the interval  ,
.
 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  t  20 minutes, is shown below.
t (min.)
R  t  (gal/min)
0
25
4
28
9
33
17
42
20
46
(a) Use data from the table and four subintervals to find a left Riemann sum to approximate the value
of
 0 R  t  dt .
20
(b) Use data from the table and four subintervals to find a right Riemann sum to approximate the value
of
 0 R  t  dt .
20
(c) A model for the rate at which water is being pumped into the tank is given by the function
W  t   25e 0.03t , where t is measured in minutes and W  t  is measured in gallons per minute.
Use the model to find the value of
20
0
W  t  dt .
CALCULUS
WORKSHEET ON ALGEBRAIC & U-SUBSTITUTION
Work the following on notebook paper. Do not use your calculator.
Evaluate.
2x

1.
 x x  2 dx
5.
2.
x
6.  x 3 x  1 dx
3.
2
 x 1  x dx
4.

x 4
2
dx
9.
7
x 2  2 dx
1
x
dx
x4
7.
 
8.
1
1
1

3
x x 2  1 dx
2
1 2 x
0
11.
4
x3  1 dx
1
dx
2x 1
4
10.
2
3
3
x
dx
x5
x
dx
2x 1
5
________________________________________________________________________________________
12. Find the area bounded by the graph of y  x 3 x  1 and the x-axis on the
interval [0, 7].
13. Find the area bounded by the graph of y  x  cos x and the x-axis on the
  
interval  ,  .
3 2
________________________________________________________________________________________
14. Solve:
dy
48

given that   1, 3 is a point on the solution curve.
3
dx
 3x  5
________________________________________________________________________________________
Given that f  x  is an even function and that
15.
  2 f  x  dx
0
16.
8
 0 f  x  dx  3 , find:
2
  2 f  x  dx
2
17.
  2 3 f  x  dx
0
________________________________________________________________________________________
Given that f  x  is an odd function and that
18.
  2 f  x  dx
0
19.
8
 0 f  x  dx  3 , find:
2
  2 f  x  dx
2
20.
  2 3 f  x  dx
0
________________________________________________________________________________________
1  1 
3
3
2
 5n 
      ...   
n  n  n 
n
 n 

21. Write lim
3

 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
n
  xk 
n 
c  x0 , x1, ..., xn where x0  x1  x2  ...  xn 1  d . Write lim
i 1
2
 x as a definite integral.
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
1. y  2  2 and y 1  6. Find y  3 .
x
 
.
4
2. f   x   cos  2 x  and f  0   3. Find f 
________________________________________________________________________________________
Work problems 3 – 7 using the Fundamental Theorem of Calculus and your calculator.
 
3. f   x   cos x3 and f  0   2. Find f 1 .
4. f   x   e x and f  5  1. Find f  2 .
2
5. A particle moving along the x-axis has position x  t  at time t with the velocity of the particle
 
v  t   5sin t 2 . 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 t  0 its velocity is given by v  t  
t
. At time
1 t2
t = 0, the position of the particle is s  0  5. 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.
y
1 f   x  dx  6.2
4
and f 1  3. Find f  4  .
x
1
9. The graph of f  is the semicircle shown on the right.
4
y
Find f   4 given that f  4  7.
x
-4
10. The graph of f  , consisting of two line segments
and a semicircle, is shown on the right. Given
that f  2   5 , find:
(a) f 1
(b) f  4 
(c) f 8
4
y
x
TURN->>>
11. Region A has an area of 1.5, and
(a)
 2 f  x  dx
(b)
 0 f  x  dx
 0 f  x  dx  3.5.
6
Find:
6
6
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 r  t   7e 0.3t 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 F  2   3 to sketch the
graph of F  x  . Label the values
of at least four points.
CALCULUS
WORKSHEET 2 ON FUNDAMENTAL THEOREM OF CALCULUS
Work these on notebook paper. Use your calculator on problems 3, 8, and 13.
1. If f 1  12, f  is continuous, and
2. If
1 f   x  dx  17, what is the value of f  4 ?
4
 2  2 f  x   3 dx  17, find  2 f  x  dx.
5
5
3. 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 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 f  0  100, estimate f  x  for x = 2, 4, 6.
Use a right Riemann sum.
x
f  x
0
10
2
18
4
23
6
25
5. Consider the function f that is continuous on the interval [ 5,5] and for which
 0 f  x  dx  4. Evaluate:
5
(a)   f  x   2  dx 
0
5
(b)
3
2
f  x  2  dx 
(c)
  5 f  x  dx (f
is even) 
(d)
  5 f  x  dx (f
is odd) 
5
5
6. Use the figure on the right and the
fact that P  0  2 to find values
of P when t = 1, 2, 3, 4, and 5.
7. Using the figure on the right, sketch
a graph of an antiderivatives G  t  of g  t 
satisfying G  0  5. Label each critical
point of G  t  with its coordinates.
TURN->>>
8. Find the value of F 1 , where F   x   e x and F  0   2.
2
2 x, x  1
9. Given f  x   
. Evaluate:
 2, x  1
 12 f  x  dx.
5
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)
Temperature T  x  (°F)
(a) Find
0
105
5
99
8
97
12
93
 0 T   x  dx .
12
(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 f 1  4, find:
y
(a) f   2
x
(b) f  5
12. (Multiple Choice) If f and g are continuous functions such that g   x   f  x  for all x ,
 f  x  dx 
3
then
2
(A) g   2  g   3
(B) g   3  g   2
(D) f  3  f  2
(E) f   3  f   2
13. (Multiple Choice) If the function
(C) g  3  g  2 
f  x  is defined by f  x   x3  2 and g is an antiderivatives
of f such that g  3  5 , then g 1 
(A)  3.268
(B)  1.585
(C) 1.732
(D) 6.585
14. (Multiple Choice) The graph of f is shown in the figure at right.
 f  x  dx  2.3 and F   x   f  x  , then F 3  F  0 
3
If
1
(A) 0.3
(B) 1.3
(C) 3.3
(D) 4.3
(E) 5.3
(E) 11.585
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 f AVE  f  c  .
1. f  x    x  3 ,  2, 5
2. f  x  
2
x , 0, 4
__________________________________________________________________________________________
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
f  x
20
42
25
38
30
31
35
29
40
35
45
48
50
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
 t 
T  t   50  14sin   . Find the average temperature during the period from 9 AM to 9 PM.
 12 
__________________________________________________________________________________________
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 T  t   20  75e
 t 50
. 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
1
 0 C  t  dt and 90  0  0 C  t  dt .
TURN->>>
y
8. Using the figure on the right,
(a) Find
1 f  x  dx .
6
(b) What is the average value of f on [1, 6]?
x
Graph of f
________________________________________________________________________________________
9. The average value of y  f  x  equals 4 for 1  x  6 and equals 5 for 6  x  8 .
What is the average value of f  x  for 1  x  8 ?
________________________________________________________________________________________
10. Suppose
 0 f  x  dx  6 .
3
(a) What is the average value of f  x  on the interval x = 0 to x = 3?
(b) If f  x  is even, what is the value of
  3 f  x  dx ?
3
What is the average value of f  x  on
the interval x   3 to x = 3?
(c) If f  x  is odd, what is the value of
  3 f  x  dx ?
3
What is the average value of f  x  on
the interval x   3 to x = 3?
________________________________________________________________________________________
In problems 11 – 14, find the average value of the function on the given interval without integrating.
Hint: Use Geometry. (No calculator)
 x  4,  4  x  1
on   4, 2
 x  2, 1  x  2
y
11. f  x   
x
12. f  x   1  1  x 2
 1, 1
y
x
13. f  x   sin x, 0, 2 
  
14. f  x   tan x,   , 
 4 4
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