Volumes of Solids of Revolution:

• Disk Method

• Washer Method

Make Sure You Remember Process for Calculating Area

Divide the region into n pieces.

Approximate the area of each piece with a rectangle.

Add together the areas of the rectangles.

Take the limit as n goes to infinity.

The result gives a definite integral.

General Idea - Slicing

1. Divide the solid into n pieces (slices).

2. Approximate the volume of each slice.

3. Add together the volumes of the slices.

4. Take the limit as n goes to infinity.

5. The result gives a definite integral.

Disk Method

Volume of a Slice

Volume of a cylinder?

h

r

2V r h

What if the ends are not circles?

A

V Ah

What if the ends are not perpendicular to the side?

No difference!(note: h is the distance between the ends)

Volume of a Solid

1

lim ( )n

kn

k

V A x x

a xk b

A(xk)

( )slice kV A x x

x

( )b

aA x dx

The hard part?

Finding A(x).

Volumes by Slicing: ExampleFind the volume of the solid of revolution formed by rotating the region bounded by the x-axis and the graph of from x=0 to x=1, about the x-axis.

y = x

Here is a Problem for You:Find the volume of the solid of revolution formed by rotating the region bounded by the x-axis and the graph of y = x4, from x=1 to x=2, about the x-axis.

Ready?

A(x) = p(x4)2= px8.

Washer Method

Setting up the Equation

Outer

FunctionInner

Function

R

r

Solids of RevolutionA solid obtained by revolving a region around a line.

When the axis of rotation is NOT a border of the region.

Creates a “pipe” and the

slice will be a washer.

Find the volume of the solid and subtract the volume of the hole.

f(x)g(x)

xk ba

NOTE: Cross-section is perpendicular to the axis of rotation.

2 2

( ) ( )b b

a aV f x dx g x dx

2 2

( ) ( )b

aV f x g x dx

Example:Find the volume of the solid formed by revolving the region bounded by y = (x) and y = x² over the interval [0, 1] about the x – axis.

2 2([ ( )] [ ( )] )

b

a

V f x g x dx

1

0

222

dxxxV

V = p (x- x4 )dx0

1

ò

V = px2

2-x5

5

æ

èç

ö

ø÷

0

1

=3

10

Here is a Problem for You:

Ready?

So……how do you calculate volumes of revolution?

• Graph your functions to create the region.

• Spin the region about the appropriate axis.

• Set up your integral.

• Integrate the function.

• Evaluate the integral.