Avon High School Name __________________________________
AP Calculus AB
Period _____
Score ______ / 10
Skill Builder: Topics 8.9-8.12 – Volumes Using Disc and Washer Methods
For each problem, sketch the region bounded by the graphs of the functions and find the area of the region.
Be sure to draw in all representative rectangles. Problems marked with an calculator icon indicate that you may
use your TI-Nspire to evaluate the definite integral.
1.) Find the volume if the region enclosing y  4  x, x  0, and y  0 is rotated about the given line.
Circle which method you will use and fill in the appropriate values of R and r where applicable.
b.) the line y  4
a.) the x-axis
Disk Method
Disk Method
r = 4 x 0
r = __________
Washer Method
Washer Method
R = __________
R=
r = 4  (4  x)
r = __________
4
4
V     4  x  dx    16  8x  x2  dx
2
0
4
4
V     42  x2  dx    16  x 2  dx
0
4

x3 
64  64

  16 x  4 x2      64  64    
3 0
3 3


c.) the line y  5
0
0
4

x 
64  128

  16 x      64   

3
3 3


0
3
d.) the y-axis
Disk Method
Disk Method
r = 4 y 0
r = __________
Washer Method
4


0
4
Washer Method
R= 5
R = __________
r = 5  (4  x)
r = __________
4
4
0
0
V    52  1  x  dx     25  1  2 x  x 2  dx     24  2 x  x 2  dx
2
40

x3 
64  176

   24 x  x2      96  16   

3 0
3 3


4
4
V     4  y  dy    16  8 y  y 2  dy
2
0
0
4

y 
64  64

  16 y  4 y 2      64  64    
3 0
3 3


3
e.) the line x  4
f.) the line x  6
Disk Method
r = __________
Disk Method
r = __________
Washer Method
Washer Method
R = 40
R = 60
r = 4  (4  y)
r = 6  (4  y)  2  y
4


4
V    62   2  y  dy     36  4  4 y  y 2  dy
2
0
0
4
4
4
V     42  y 2  dy    16  y 2  dy
0

y3 
64  224

   32 y  2 y 2     128  32   

3 0
3
3


0
4

y3 
64  128

  16 y      64   

3 0
3 3


2.) Find the volume if the region enclosing y  x2  x, y  0, and x  2 is rotated about the given line.
Circle which method you will use and fill in the appropriate values of R and r where applicable.
b.) the line y  6
a.) the x-axis
Disk Method
Disk Method
2
r= x x
r = __________
Washer Method
Washer Method
R = __________
R = 60
r = __________
r = 6  ( x 2  x)
2
2
2
V     x2  x  dx     x 4  2 x3  x 2  dx
2
0
0
2
 x5 x 4 x3 
8  256
 32
       8  

3  15
 5
 5 2 3 0


2


V    (6  0)2   6  ( x 2  x)  dx    36   6  x 2  x  dx 
0
2
2
     x 4  2 x3  11x 2  12 x  dx
0
2
 x5 x 4 11x3

584
   
 6 x2  

5
2
3

 0 15
0
2
c.) the line y  9
2


2


V    (9  0)2   9  ( x 2  x)  dx    81   9  x 2  x  dx
Disk Method
r = __________
0
2
0
2
2
     x 4  2 x3  17 x 2  18x  dx
0
2
Washer Method
 x5 x 4 17 x3

1004
   
 9 x2  

3
15
 5 2
0
R = 90
r = 9  ( x 2  x)
3.) Find the volume if the region enclosing y  x3 , x  0, and y  8 is rotated about the given line.
Circle which method you will use and fill in the appropriate values of R and r where applicable.
b.) the line y  8
a.) the x-axis
Disk Method
Disk Method
r = __________
r = 8  x3
Washer Method
2

R = 80
R = __________
r = x3  0
r = __________

2
V    (8  0)2   x3  0  dx     64  x6  dx
0
2
0
2

x 
128  768

   64 x     128 

7
7 
7


0
7
Washer Method
c.) the line y  9
2
2
V    8  x3  dx     64  16 x3  x6  dx
2
0
0
2

x7 
128  576

   64 x  4 x4     128  64 


7
7  7


0
d.) the y-axis
Disk Method
Disk Method
r = __________
r=
Washer Method
2


0
r = 9 8
r = __________
2
2

9x x 
128  744

   80 x 
    160  72 

2
7
7 
7


0
7
Washer Method
R = __________
0
4
y 0
R = 9  x3
V    (9  x3 )2   9  8 dx    80  18x3  x6  dx
2
3
8
8
V     y1/3  0  dx     y 2/3  dx
2
0
0
8
3

3
 96
   y5/3      32   
5
0
5
 5
e.) the line x  2
f.) the line x  3
Disk Method
8


r = __________
r = __________
Washer Method
Washer Method
R = 20
R = 30
r = 2 3 y
r = 3 3 y
8
V    (2  0)2   2  y1/3  dy     4  4  4 y1/3  y 2/3  dx
0
2
Disk Method
0
3
96  144
 3


   4  y 4/3  y5/3     48   
4
5
5
5

0

8
8


8
V    (3  0)2   3  y1/3  dy     9  9  6 y1/3  y 2/3  dx
0
2
0
3
96  264
 3


   6  y 4/3  y5/3     72   
4
5
5
5

0

8
4.) Find the volume if the region enclosing y  x 2 and y  2 x where x  0 is rotated about the given
line. Circle which method you will use and fill in the appropriate values of R and r where
applicable.
a.) the x-axis
b.) the line y  4
Disk Method
Disk Method
r = __________
r = __________
Washer Method
Washer Method
R = 2x  0
R = 4  x2
r = x2  0
r = 4  2x
2
2


2
V    (2 x  0)   x  0  dx     4 x  x  dx
0
2
2
2
2
2
0
 x x 
 32 32  64
   4        
3
5
 3 5  15

0
3
5
4


V    (4  x 2 )2   4  2 x  dx 
0

32
5
2
c.) the line y  7
d.) the y-axis
Disk Method
Disk Method
r = __________
r = __________
Washer Method
Washer Method
R = 7  x2
r = 7  2x
2

r=

V    (7  x 2 )2   7  2 x  dx 
2
0
y  2x  x 
72

5
y 0
R=
y
; y  x2  x  y
2
4

V 
 


y
0
2

2
y

y  0    0
2

2

 dy

0
8

3
e.) the line x  2
f.) the line x  3
Disk Method
Disk Method
r = __________
r = __________
Washer Method
Washer Method
R = 3
y
R = 2
2
r = 3
r = 2 y
y  2x  x 
y
; y  x2  x  y
2
4
2
2
 
y
V 
2

 2  y  dy


  

2



0

8
3

y  2x  x 
y
; y  x2  x  y
2
4
2
2
 
y
V 
3

 3  y  dy



 

2




0
16

3

y
2
y
5.) Find the volume if the region enclosing y  1 x , x  0, y  0 and x  9 is rotated about the given
line. Circle which method you will use and fill in the appropriate values of R and r where
applicable.
a.) the x-axis
b.) the line y  4
Disk Method
Disk Method
r = 1 x
r = __________
Washer Method
Washer Method
R = __________
R = 40
r = __________
9



r = 4  1 x
2
V    1  x dx
2
V     4  0   4  1  x

0
261

2
0

 
9
171
2
c.) the line y  5
  dx
2
d.) the y-axis This requires both disk and washer methods.
Disk Method
Disk Method
r = 90
r = __________
Washer Method
Washer Method
R = 90
R = 50

r = 5  1 x
9
 
2
V     5  0   5  1  x

0
369

2

  dx
2
r = ( y  1)2  0
y  1  x  x  y  1  x  ( y  1)2
1
4
2
V  V1  V2    (9  0)2 dy    (9  0)2   ( y  1) 2  0   dy


0
1

e.) the line x  9

This requires two disk methods.
Disk Method
r1 = 9  0
r2 = 9  ( y  1)2
Washer Method
R = __________
r = __________
1377
5
y  1  x  x  y  1  x  ( y  1)2
1
4
0
1
V  V1  V2    (9  0)2 dy     9  ( y  1)2  dy
1053

5
2
 
6.) Find the volume if the first quadrant region enclosing y  sin x and y  cos x on 0,  is rotated
 4
about the given line. Circle which method you will use and fill in the appropriate values of R and r
where applicable.
a.) the x-axis
b.) the y-axis This requires two disk methods.
Disk Method
Disk Method
r1 = arcsin y  0
r = __________
r2 = arccos y  0
Washer Method

V 
R = cos x  0
Washer Method
r = sin x  0
R = __________
r = __________
4
  cos x  0  sin x  0  dx
2
2
1
0

V  V1  V2  

2
0
2
1
2
2
  arcsin y  0 dy   1  arccos y  0 dy
2
 0.221
Note: cos x  sin x  cos 2 x. This would be fairly
easy to integrate.
2
2
7.) A tank on the wing of a jet plane is formed by revolving the region bounded by the graph of
1
y  x2 3  x and the x-axis about the x-axis where x and y are measured in meters. Find the volume
10
of the tank. A calculator may be needed to simplify the definite integral’s result.
3
 1

V     x 2 3  x  dx

  10
2
0


3
4
  x (3  x)  dx 
100 0
3
  3x5
x6 

 

100  5
6 0

  729 2187 
100  5
243

1000

6 

3
 3x
100 
0
4
 x5  dx
8.) The region bounded by the curve y  x , x  0, y  0 and x  9 is rotated about the x-axis.
a.) Find the value of a in the interval [0,9] that divides the region into 2 parts of equal area.
Write your answer as both an exact value and as a decimal approximation using your calculator.
9
A
 
9
2
2
x dx  x3/2   27  18
3
3
0
0
  x  dx  9
a
0
a
2 3/2
x
9
3
0
2 3/2
a 9
3
27
a3/2 
2
2/3
 27 
a     5.6696
 2 
b.) Find the value of a in the interval [0,9] that divides the region into 2 parts of equal volume.
Write your answer as both an exact value and as a decimal approximation using your calculator.
9
V 
 
0
a

0
 x  dx  814
2
 x2
2
a
a

0
81
4
81
2
4
81
a2 
2
9
a
 6.3640
2
2

9
x2
81
x dx    x dx   

2 0
2
0
2
9