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.
Slice R r rotate around x axis dt More Volumes f(x) g(x) Area of a slice = R2 – r2
Slice Big R little r R r Volumes by Washers f(x) g(x) f(x) 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