CALCULUS BC
WORKSHEET ON 7.1
Work the following on notebook paper.
Find the area bounded by the given curves. Draw and label a figure for each problem, and show all work. Do not
use your calculator on problems 1 - 2.
1. f  x   x 2  2 x  1, g  x   3x  3
2. x  y 2 , x  y  2
____________________________________________________________________________________________
Find the area bounded by the given curves. Draw and label a figure for each problem, set up the integral(s) needed,
and then evaluate on your calculator.
y
3. f  x   x 4 , g  x   3x  4
4. f  x   x 2 , g  x   2 x (see figure on the right)


5. y  ln x 2  1 , y  cos x
x
Figure for prob. 4
____________________________________________________________________________________________
6. (No calculator)
Let R be the region in the first quadrant bounded by the
(4, 2)
x-axis and the graphs of y  x and y  6  x as shown
in the figure on the right.
(a) Find the area of R by working in x’s.
R
(b) Find the area of R by working in y’s.
(c) When you found your answers to (a) and (b), was it less work to work in terms of x or in terms of y?
____________________________________________________________________________________________
7. (Calculator)
Let R be the region bounded by the graphs of
1
R
y  2 x , y  4 x , and y  , as shown in the figure
x
on the right. Find the area of R
Answers to Worksheet on 7.1
1.
9
2
  3x  3   x  2 x  1 dx  ...  2
2.
9
   y  2  y  dy  ...  2
2
1
2
2
1
3. x4  3x  4 at x  1.742... Let a  1.742...
4
   3x  4  x  dx  10.612
a
1
4. x  2x at x   0.766..., x  2, and x  4. Let a   0.766...
2
 2
2
a

x

 x 2 dx  
4
2

 x2  2  dx  3.460
x
5. ln x 2  1  cos x at x   0.915... and x  0.915... Let a  0.915...

a
a
 cos x  ln  x  1 dx  1.168
6. (a) Area =
(b) Area =
2

4
0
x dx    6  x  dx  ... 
6
4
2
  6  y  y  dy  ... 
2
0
22
3
22
3
(c) Working in terms of y was less work.
1
1
1
at x  , and 2 x 
at x  0.641.... Let a  0.641...
x
2
x
1
a 1
2
x
x
x
0  4  2  dx  12  x  2  dx  0.163
7. 4 x 
CALCULUS BC
WORKSHEET ON VOLUME BY CROSS SECTIONS
Work the following problems on notebook paper. For each problem, draw a figure, set up an integral, and then
evaluate on your calculator. Give decimal answers correct to three decimal places.
1. Find the volume of the solid whose base is bounded by the graphs of y  x  1 and y  x 2  1 , with the
indicated cross sections taken perpendicular to the x-axis.
(a) Squares
(b) Rectangles of height 1
(c) Semiellipses of height 2 (The area of an ellipse is given by the formula A   ab , where a and b
are the distances from the center to the ellipse to the endpoints of the axes of the ellipse.)
(d) Equilateral triangles
2. Find the volume of the solid whose base is bounded by the circle x 2  y 2  4 with the indicated cross
sections taken perpendicular to the x-axis.
(a) Squares
(b) Equilateral triangles
(c) Semicircles
(d) Isosceles right triangles with the hypotenuse as the base of the solid
3. The base of a solid is bounded by y  x3 , y  0, and x  1. Find the volume of the solid for each of the
following cross sections taken perpendicular to the y-axis.
(a) Squares
(b) Semicircles
(c) Equilateral triangles
(d) Semiellipses whose heights are twice the lengths of their bases
Answers to Worksheet on Volume by Cross Sections

(b) V     x 2  x  2  dx  4.5

9
(c) V     x 2  x  2  dx  7.069 or
2
4
1. (a) V 
 1   x
2
2
2
 x  2 dx  8.1
2
1
2
2
1
(d) V 


2
3 2
81 3
 x 2  x  2 dx  3.529 or


1
4
40
2
  2  4  x  dx  42.667 or
128
3
2
32 3
(b) V  3 
4  x 2 dx  18.475 or
2
3
2

16
(c) V  
4  x 2 dx  16.755 or

2
2
3
2
32
(d) V  
4  x 2 dx  10.667 or
2
3
2. (a) V  4
2





3. (a) V 

 0 1  3 y 
1
dy  0.1

1  y  dy  0.039 or


8
80
3
3
(c) V 
1  y  dy  0.043 or


4
40


(d) V   1  y  dy  0.157 or
2
20
(b) V 

2
1
2
3
0
1
2
3
0
1
0
3
2
CALCULUS BC
WORKSHEET ON 7.2 – VOLUME BY DISCS & WASHERS
Work the following on notebook paper.
Do not use your calculator on problems 1 – 3. Draw a figure and shade the enclosed region. Then find the volume
of the solid generated by revolving the given region about the given axis.
1. y 
x , y  0, x  4 about the x-axis
2. y  4  x 2 , y  0, and the y  axis about the x-axis
3. y  x 2 and y  x3 about the x-axis
_________________________________________________________________________________
Use your calculator on problems 4 – 6. Draw a figure and shade the enclosed region. Then find the volume of the
solid generated by revolving the given region about the given axis by setting up an integral expression and then
using your calculator. Give your answers correct to three decimal places.
4. y  x , y  0, and x  4
(a) about the line y  3
(b) about the line y   2
5. y  x 2 and y  4 x  x 2
(a) about the x-axis
(b) about the line y  6
(c) about the line y  1
6. y  x and y  x 2
(a) about the x-axis
(b) about the line y  3
(c) about the line y   2
Answers to Worksheet on Volume by Discs and Washers
1. V  
  x  dx  ...  8
2
4
0
512
2
  4  x  dx  ...  15
2
3. V      x 2    x3  dx  ... 
35
2. V  
2
2
2
1
2
2
0
3  3  x   dx  75.398 or 24
88
V     x  2   2  dx  92.153 or
3
4. (a) V  

4
2
2
0
2
4
(b)
2
0
5. x 2  4 x  x 2 at x  0 and x  2
(a) V  
(b) V  
(c) V  
6.
2
2
  4 x  x    x   dx  33.510
2
2
2
0
or
2
2
  6  x    6  4 x  x   dx  67.021 or
2
2
2
0
64
3
2
2
  4 x  x  1   x  1  dx  50.265 or 16
2
2
2
0
x  x at x  0 and x  1
2
  x    x   dx  0.942
(b) V     3  x    3  x   dx  5.341
(c) V     x  2    x  2   dx  5.131
(a) V  
32
3
1
2
2 2
0
1
2
2 2
0
1
0
2
2
2
CALCULUS BC
WORKSHEET ON 7.2
Work the following on notebook paper.
On problems 1 – 2, use your calculator, and give your answers correct to three decimal places. Each answer must
have supporting work.
1. Let R be the region enclosed by the graph of y  x  1 , the vertical line x = 10, and the x-axis.
(a) Find the volume of the solid generated when R is revolved about the horizontal line y = 3.
(b) Find the volume of the solid generated when R is revolved about the horizontal line y   2.
(c) The region R is the base of a solid. For this solid, the cross sections perpendicular to the x-axis are
x2 3
equilateral triangles. Find the volume of this solid. (The area of an equilateral triangle is A 
4
where x is the length of a side.)
2. Let R be the shaded region bounded by the graphs of y  x and y  e3 x and the vertical
line x = 1, as shown.
(a) Find the volume of the solid generated when R is revolved about the
horizontal line y   2.
(b) Find the volume of the solid generated when R is revolved about the
horizontal line y = 1.
(c) The region R is the base of a solid. For this solid, each cross section
perpendicular to the x-axis is a rectangle whose height is 5 times the length
of its base in region R. Find the volume of this solid.
_____________________________________________________________________________________
Do not use your calculator on problems 3 – 4.
3. Let R be the region enclosed by the graphs of y  x 2 and y  12  x.
(a) The region R is the base of a solid. For this solid, the cross sections perpendicular to the x-axis are
squares. Write, but do not evaluate, an integral expression that gives the volume of this solid.
(b) The region R is revolved about the x-axis. Write, but do not evaluate, an integral expression that
gives the volume of the solid formed.
(c) The region R is revolved about the horizontal line y = 16. Write, but do not evaluate, an integral
expression that gives the volume of the solid formed.
4. Let R be the region enclosed by the graphs of y 
1
, y  x 2 , and y  4.
x
(a) The region R is the base of a solid. For this solid, the cross sections
perpendicular to the y-axis are semicircles. Write, but do not evaluate,
an integral expression that gives the volume of this solid.
(b) The region R is revolved about the vertical line x   3. Write, but do not
evaluate, an integral expression that gives the volume of the solid formed.
R
Answers to Worksheet on 7.1 – 7.2
1. (a) A  
10
(b) V  

10
1


32  3  x  1
  dx  212.058 or 1352
2
3 10
81 3
or 17.537
 x  1 dx 

1
4
8
(c) V 
2.
x  1 dx  18
1
x  e 3x at x  0.2387...
(a) A  
1
a
(b) V  


x  e  3 x dx  0.443
 1  e   1 
1
 3x 2
a

1
(c) V  5
Let a  0.2387...
x  e 3x
a
x
  dx  1.424 or 0.453
2
 dx  1.554
2
3. x  12  x at x   4 and x  3
3
599
(a) A   12  x  x 2 dx  ... 
4
6
2


(b) V   12  x  x 2  dx
(c) V    12  x    x 2   dx
3
2
4
3
2
2
4
4
1
14
4. (a) A    y   dy  ...   ln 4
1
y
3

4
(b) V 
 
2
1
  y   dy
8 1 
y

(c) V   

1
4





1

y  3    3
y


2
2

 dy


CALCULUS BC
WORKSHEET ON 7.1 – 7.2
Do not use your calculator.roblems 1 - 3. Draw a figure for each problem, and show your supporting work.
1. Let R be the region bounded by the graphs of y  2 x and y  x 2 .
(a) Find the area of the region R.
(b) Find the volume of the solid generated when R is revolved about the x-axis.
(c) Find the volume of the solid generated when R is revolved about the horizontal line y = 5.
(d) The region R is revolved about the vertical line x = 5. Write, but do not evaluate, an integral expression
that gives the volume of the solid.
(e) The region R is the base of a solid. For this solid, each cross section perpendicular to the x-axis is a
semicircle. Write, but do not evaluate, an integral expression that gives the volume of the solid.
(f) The region R is the base of a solid. For this solid, each cross section perpendicular to the y-axis is a square.
Write, but do not evaluate, an integral expression that gives the volume of the solid.
2. Let R be the region bounded by the graphs of y 
x
x and y  .
3
(a) Find the area of the region R.
(b) Find the volume of the solid generated when R is revolved about the x-axis.
(c) The region R is revolved about the horizontal line y  1. Write, but do not evaluate, an integral expression
that gives the volume of the solid.
(d) The region R is revolved about the vertical line x  1. Write, but do not evaluate, an integral expression
that gives the volume of the solid.
(e) The region R is the base of a solid. For this solid, each cross section perpendicular to the x-axis
is an equilateral triangle. Write, but do not evaluate, an integral expression that gives the volume
of the solid.
(f) The region R is the base of a solid. For this solid, each cross section perpendicular to the y-axis
is a rectangle whose height is three times the length of the base. Write, but do not evaluate, an
integral expression that gives the volume of the solid.
3. Let R be the region bounded by the graphs of x  16  y 2 and x  4  y . Find the area of R.
___________________________________________________________________________________________
You may use your calculator on problem 4.
4. Let R be the region bounded by the graphs of
f  x   20  x  x2 and g  x   x 2  5 x.
(a) Find the area of R.
R
(b) A vertical line x = k divides R into two regions of equal area.
Write, but do not solve, an equation that could be solved to
find the value of k.
(c) The region R is the base of a solid. For this solid, the cross
sections perpendicular to the x-axis are isosceles right triangles
Graph for Problem 4
with the hypotenuse as the base of the solid. Write, but do not
evaluate, an integral expression for the volume of this solid.
(d) Write, but do not evaluate, an integral expression for the volume of the solid generated when R is rotated
about the horizontal line y  10.
Answers to Worksheet on 7.1 – 7.2
4
3
64
(b)
15
136
(c)
15
1. (a)
4

(d)  

0
(e)

2

y
  5    5  y
2



  dy
2

2 x  x 2  dx


8
2
2
0
4
 
y
(f)   y   dy
2
0 
2
9
2
27
(b)
2
2. (a)
9
 
(c)   
 
0 


x 
x  1    1
3 
2
2

 dx

 3 y  1   y  1  dy
 

(d)  
3
2
2
2
0
9
3 
x
(e)
 x   dx

4 0 
3
2
2
  3y  y  dy
(f) 3
3
2
0
343
6
343
4. (a)
3
3.

2
2
  20  x  x    x  5x   dx  2 
k
(b)
2
2
 10  3x  x  dx
5
(c)
2
2
(d)  

1 343 

3 
5
 2
30  x  x2    x2  5x 10  dx
2
2
CALCULUS
WORKSHEET ON AREA AND VOLUME
Work the following on notebook paper. Do not use your calculator.
1. Let R be the region bounded by the graphs of y  4  x 2 and y  x  2.
(a) Find the area of R.
(b) The region R is the base of a solid. For this solid, the cross sections perpendicular to the x-axis are squares.
Write, but do not evaluate, an integral expression for the volume of this solid.
(c) Write, but do not evaluate, an integral expression for the volume of the solid generated when R is rotated
about the horizontal line y = 6.
(d) Write, but do not evaluate, an integral expression for the volume of the solid generated when R is rotated
about the vertical line x   3.
2. Let R be the region in the first quadrant bounded by the graphs
of y  2 x , the horizontal line y = 6, and the y-axis, as shown
in the figure on the right.
(a) Find the area of R.
(b) Write, but do not evaluate, an integral expression for the volume
of the solid generated when R is rotated about the vertical line
x = 12.
(c) Write, but do not evaluate, an integral expression for the volume
of the solid generated when R is rotated about the horizontal line
y = 7.
(d) Region R is the base of a solid. For each y, where 0  y  6, the cross section of the solid taken
perpendicular to the y-axis is a rectangle whose height is 3 times the length of its base in region R.
Write, but do not evaluate, an integral expression that gives the volume of the solid.
3. Let R be the region bounded by the x-axis, the graph of y  x , and the line x = 4.
(a) Find the area of the region R.
(b) Find the value of h such that the vertical line x = h divides the region R into two regions of equal area.
(c) Find the volume of the solid generated when R is revolved about the x-axis.
(d) The vertical line x = k divides the region R into two regions such that when these two regions are revolved
about the x-axis, they generate solids with equal volumes. Find the value of k.
4. Let R be the region in the first quadrant bounded by the graphs of y  x, y 
1
x2
, the x-axis and the
vertical line x = 3.
(a) Find the area of the region R.
(b) The region R is the base of a solid. For this solid, the cross sections perpendicular to the x-axis are
rectangles with height five times the length of the base. Find the volume of this solid.
(c) Write, but do not evaluate, an integral expression for the volume of the solid generated when R is
rotated about the horizontal line y = 2.
Answers to Worksheet on Area and Volume
1. (a)
9
2
2
  2  x  x  dx
(c)  
   4  x    2  x 2   dx

(b)
1
2
2
1
2
2
2
2
  x  3  2  x  x  dx
1
(d) 2
2
2. (a) 18
 12  x   6  2 x  dx
9
(b) 2
0
7  2
 
9
(c)  

0
x

2

 12 dx
6
3y4

dy
(d) 
 0 16
16
3. (a)
3
2
(b) 4 3
(c) 8
(d) 2 2
7
6
265
(b)
81
4. (a)
(c) 
  2   2  x 
1
0
2
2

3
2
  2 
1  
dx      2    2  2   dx
 
x  

1 
CALCULUS BC
WORKSHEET ON ARC LENGTH AND REVIEW
Work the following on notebook paper.
On problems 1 – 3, find the arc length of the graph of the function over the indicated interval. Do not use your
calculator on problems 1 - 3.
2 3
3 2
  3 
1. y  x 2  1,  0, 1
2. y  x 3 , 1, 8
3. y  ln  sin x  ,
 4 , 4 
3
2
__________________________________________________________________________________________
On problems 4 – 7, sketch the graph of the given function, set up an integral that represents the arc length of the
graph of the function over the indicated interval, and evaluate it on your calculator. Give your answers correct to
three decimal places.
4. y  4  x 2 , 0  x  2
6. y  sin x, 0  x  
1
5. y  , 1  x  3
7. x  e y , 0  y  2
x
__________________________________________________________________________________________
On problems 8 – 9, sketch the graphs of the given functions, and find the area bounded by the graphs of the
functions. Do not use your calculator.
2
8. f  x    x  4 x  1, g  x   x  1
2
9. f  y   y , g  y   y  6
__________________________________________________________________________________________
10. (Modified version of 2009 AB 4) (No calculator)
Let R be the region in the first quadrant enclosed by the graphs of y  2 x and y  x 2 .
(a) Find the area of R.
(b) The region R is the base of a solid. For this solid, at each x, the cross section perpendicular to the x-axis
 
has area A  x   sin  x  . Find the volume of the solid.
2 
(c) Another solid has the same base R. For this solid, the cross sections perpendicular to the y-axis are squares.
Write, but do not evaluate, an integral expression for the volume of the solid.
(d) Write, but do not evaluate, an integral expression for the volume of the solid generated when R is rotated
about the horizontal line y   3.
(e) Write, but do not evaluate, an integral expression for the volume of the solid generated when R is rotated
about the vertical line x  3.
11. (2001 AB 1) (Use your calculator.)
Let R and S be the regions in the first quadrant shown
in the figure. The region R is bounded by the x-axis
and the graphs of y  2  x3 and y  tan x . The region
S is bounded by the y-axis and the graphs of y  2  x3
and y  tan x .
(a) Find the area of R.
(b) Find the area of S.
(c) Find the volume of the solid generated when S is revolved about
the x-axis.
(d) The region S is the base of a solid. For this solid, each cross section perpendicular to the x-axis is
a semicircle. Find the volume of this solid.
Answers to Worksheet on Arc Length and Review
1.


2 32
2 1
3
2. 5
3
2
2
3
2
 2 1 

 2 1 
3. ln 

4. 4.647
5. 2.147
6. 3.820
7. 2.221
9
2
125
9.
6
8.
10. (a)
4
3
(b)
4

4
 
y
(c)   y   dy
2
0 
2
 2 x  3   x  3  dx
 
(d)  

2
2
2
2
0
(e) 2
2
  3  x   2 x  x  dx
2
0
11. (a) 0.729
(b) 1.161
(c) 8.332
(d) 0.694
CALCULUS
WORKSHEET ON APPLICATIONS OF THE DEFINITE INTEGRAL - ACCUMULATION
Work the following on notebook paper. Use your calculator on problems 1 - 8 and give decimal answers correct
to three decimal places.
1. A tire factory is located at the center of a town, which extends three miles either side of the factory. Let x
be the distance from the factory. A straight highway goes through town, and the density of particles of


pollutant along the highway is given by p  x   500 12  x 2 where p  x  is the number of particles of
pollutant per mile per day at the point x. Find the total number of particles of pollutant per day deposited
in town along the highway.
2. The density function for the population of the small coastal town of Westport, WA, is given
p  x   2000  2  x  where x is the distance along a straight highway from the ocean and p  x  is measured
in people per mile. The town extends for two miles from the ocean. Find the population of Westport.
3. A city in the shape of a circle of radius 10 miles is growing
with its population density a function of the distance from the
center of the city. At a distance of r miles from the city center,
its population density is P  r  
5000
people/square mile. In
1 r
the figure, thin rings have been drawn around the center of the city.
The area in square miles of the shaded ring can be approximated by
the product 2 r  r , where r is the radius of the outer ring.
(a) Write an expression to represent the number of people in the
shaded ring.
(b) Find the population of the city.
4. Greater Seattle can be approximated by a semicircle of radius 5 miles with its center on the shoreline of Puget
Sound. Moving away from the center along a radius, the population density can be approximated by
p  x   20, 000e 0.13 x , where p  x  is measured in people per square mile. Find the population of the city.
5. The density of oil in a circular oil slick on the surface of the ocean at a distance r meters from the center
50
of the slick is given by f  r  
where f  r  is measured in kg/m 2 and r is measured in meters. If
1 r
the slick extends from r = 0 to r = 1000 m, find the exact value of the mass of oil in the slick.
 0.12t
6. Water is pumped out of a holding tank at a rate of 5  5e
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?
7. A cup of coffee at 90° C is put into a 20° C room when t = 0. The coffee’s temperature is decreasing at a rate
 0.1t
of r  t   7e
° C per minute, with t in minutes.
(a) Find the coffee’s temperature when t = 10.
(b) Find the average rate of change of the temperature of the coffee over the first ten minutes.
TURN->>>
Use your calculator on problem 8.
y
8. Let R be the region bounded by the y-axis and the graphs of
x3
y  3  2 x and y  2
as shown in the figure.
x 1
(a) Find the area of R.
R
(b) Find the volume of the solid generated when R is revolved about the
horizontal line y  1 .
x
(c) Find the volume of the solid generated when R is revolved about the
vertical line x  1 .
(d) The region R is the base of a solid. For this solid, each cross section perpendicular to the x-axis is a
square. Find the volume of this solid.
_______________________________________________________________________________________
Do not use your calculator on problem 9.
9. Let R be the region in the first quadrant bounded by the
(4, 2)
x-axis and the graphs of y  x and y  6  x as shown
in the figure on the right.
R
(a) Find the area of R.
(b) The region R is the base of a solid. For this solid, the cross
sections perpendicular to the x-axis are rectangles with a
height of 5. Wrote, but do not evaluate, an integral expression
or the volume of the solid.
(c) The region R is the base of a solid. For this solid, the cross sections perpendicular to
the y-axis are semicircles. Wrote, but do not evaluate, an integral expression for the volume
of the solid.
(d) Write, but do not evaluate, an integral expression for the perimeter of R.
Answers to Worksheet on Applications of the Definite Integral – Accumulation
1. 27,000 particles
2. 4000 people
 5000 
   2 r    r 
 1 r 
3. (a) P  r    2 r    r   
(b) 238,827 people
4. 515,387 people
5. 311,988,816 kg
6. 741.636 liters
7. (a) 45.752°C
(b) – 4.425°C per minute
8. (a) 1.888
(b) 25.516
(c) 16.626
(d) 3.948
9. (a)
(b)
22
3

4
0
5 x dx   5  6  x  dx
6
4
(c)

6  y  y 2  dy


8
2
0
4
2

6
 1 
2
(d)  1  
dx

1   1 dx  6



4
2 x 
0