Math 215, Calculus II: Computing the volumes of solids, addendum Solids of revolution: The philosophy If you want to rotate a region R about a vertical (or horizontal) axis. (1) Break the region into horizontal (or vertical, if you are using a horizontal axis) line segments (2) For each of these line segments find the distance from the endpoint closest to the axis of revolution to the axis of revolution. This is the inner radius of the washer rin(y) if you are using horizontal segments, or rin(x) if you are using vertical segments. (3) For each of these line segments find the distance from the endpoint furthest to the axis of revolution to the axis of revolution. This is the outer radius of the washer rout(y) if you are using horizontal segments, or rout(x) if you are using vertical segments. 2 2 2 (4) The area of a washer is given by A(y) = π(rout(y) −rin(y) ) Or by A(x) = π(rout(x) − 2 rin(x) ) (5) Integrate this: Z b Z b 2 2 2 2 Volume = π(rout(y) − rin(y) )dy OR π(rout(x) − rin(x) )dx a a

√ example Here is the region under the curve y = x (OR x = y2) where x runs from 0 to 4. Imagine revolving this region about the y-axis. Sketch a picture. Let’s compute the volume. Draw a generic horizontal cross section at some y-value (since the axis of revolution is vertical) What are its endpoints in terms of y? What is the distance from each of the endpoints to the axis of revolution?

rin(y) =

rout(y) =

What is the y-interval of interest?

Z b 2 2 Compute the integral π(rout(y) − rin(y) )dx. a

1 2

Suppose that you want to rotate the region bounded by y = x, y = 0 and x = 1 about the line y = 1. Since we are rotating about a horizontal line, we will consider a generic vertical line in the region. What shape do you get when you rotate it? What is the radius of the inner circle of the washer?

What is the radius of the outer circle of the washer?

The volume of this solid of revolution is given by Z b 2 2 π(rout − rin) =. a