of Solids of Revolution Volume of Solids of Revolution

How Do You Get a Solid? Start with a function Identify a region of area Rotate the region around an axis of rotation Poof! You’ve got a solid!

Look Closely at the Solid If you imagine slicing the solid perpendicular to the x-axis, what shape are the slices? Circles, of course! How would you go about finding the volume of the solid?

How Do You Find Volume? In general, how do you find the volume of a solid? Start by finding the area of one slice. Since the slices are circular, what do we need to know in order to find the area of a slice? The radius, of course!

So What’s the Radius? Is the radius a constant length? No…it changes depending on where you are on the x-axis. So how do you represent the radius if it changes? Doesn’t the radius always equal the height of the function? r = x2 - 4x + 5

So What Now? If r = x2 - 4x + 5 then the area of one slice is… A = πr2 A = π(x2 - 4x + 5)2 Now that we have one slice, how do we add up all the slices? Remember what an integral does? You know it! It adds things up!

Okay…So Let’s Integrate! Let’s say we want the volume of the solid between x = 1 and x = 4. We need to add up the slices where 1 ≤ x ≤ 4

Let’s Review… To find the volume of a solid with circular slices, start by finding the area of one slice A = πr2 Use an integral to “add up” all the slices on a given interval. Now it’s your turn to try one!

You Try: Find the volume of the solid obtained by rotating the region bounded by the function y = x2, x = 1, x = 2, and the x-axis about the x-axis. r = x2 A = πr2 = π(x2)2 = πx4

Can You Write a General Formula? Using a ≤ x ≤ b as the interval, write a general formula for finding the volume of a solid with circular slices with radius r. Basically it’s the area of one slice (πr2) integrated over the interval.

More Volumes R Slice f(x) r g(x) rotate around x axis Area of a slice = R2 – r2 dt

V =  (R2 – r2) dx Volumes by Washers f(x) R f(x) Slice g(x) Big R little r f(x) R r g(x) R = f(x) r = g(x) Thus, A = (R2 – r2) V =  (R2 – r2) dx

General Formulas: Volume =  A Disks =  r2 Washers =  (R2 – r2 )

You Try: Find the volume of the solid obtained by rotating the region bounded by the functions y = x2, y = 1.5x, x = 0, x = 1.5 about the x-axis. R = 1.5x r = x2