04/19/23 Perkins

AP Calculus AB

Day 8Section 7.3

Shell Method (for finding the volume of a solid of revolution)

2b

figure

a

V r x f x dx

radius length thickness2cylinderV

axis to rectangle length of rectangle width of rectangle2

A B

f x f y

C

D

r y

r x

2d

figure

c

V r y f y dy

Most important difference is that our rectangles are chosen to be parallel to the axis of revolution!

Vertical axis of revolution Horizontal axis of revolution

1. Use the shell method to find the volume of the solid generated by revolving the area enclosed by over about the y-axis.

3y x x

x

1

3

0

2V x x x dx

1

2 4

0

2 x x dx

Shell Method:

0 1x

1

0.5

1

3y x x

13 51 13 5 0

2 x x

1 13 52 0

4

15

2. Use the shell method to find the volume of the solid generated by revolving the area enclosed by over about the x-axis.

2yx e

y

21

0

2 yV y e dy Shell Method:

0 1y

2yx e

1

0.5

1

1

0

yu

y

e du

2u y2du ydy

1 1e

1

0

yu

ye

11

e

.6321

1.986

Perkins

AP Calculus AB

Day 8Section 7.3

Shell Method (for finding the volume of a solid of revolution)

A B

f x f y

C

D

Vertical axis of revolution Horizontal axis of revolution

1. Use the shell method to find the volume of the solid generated by revolving the area enclosed by over about the y-axis.

3y x x 0 1x

2. Use the shell method to find the volume of the solid generated by revolving the area enclosed by over about the x-axis.

2yx e0 1y