1. Volume - The Disk Method
Lesson 7.2:
Volume - The Disk Method
Objective: To find the volume of a solid with curved surfaces.

2. First, we can start with our "representative rectangle" from area.
However, we will be rotating this rectangle about the x-axis
We create a "disk" - which is a cylinder. We can find the area of a cylinder.
V=πr2h
To find the overall volume we will need calculus to find the volume of all the infinitely small, individual cylinders

3. Q: So how do we find the volume of the whole region?
A: Creating an infinite number of cylinders that represent
the entire solid.

4. We could put any solid on a graph to get an equation

5. Let's see the progression:

6. So let's change our equation for the entire solid where the curved boundary of the solid is defined by a function R(x)
V = π r2 h
What represents the radius of the disk?
What determines the height of the disk?
ΔV = π (R(x))2 Δx
Now, find the volume as a limit of infinitely many disks.

7. Remember from area, we can calculate this rotating with respect to the x or y

9. You need to be very careful about how you think about the radius!
Problem Set 7.2.1

10. The Washer Method Occurs when there is a "hole" in the solid
The volume equation then becomes
Let
R(x) = outer radius, and
r(x) = inner radius

12. Using Two Integrals With Respect to Y

13. Volume of Other Solids - With Known Cross Sections
The method using cylinders can also apply the same method with other cross sections
We can calculate volume as long as we have a formula for the area of the cross section
Across-section = side2
So we can write two equations:
Horizontal rotation:
Vertical rotation: