1. AP Calculus BC April 3-4, 2017 Mrs. Agnew
Volumes of Revolution
AP Calculus BC
April 3-4, 2017
Mrs. Agnew

2. Essential Question Essential Vocabulary
How do find the volumes of solids using the washer method and the disk method?
Essential Vocabulary
Volumes of Revolution
Washer Method
Disk Method

3. Find the Given Volumes
Find the volume of the solid formed when f(x) is rotated about the given axis
y = x axis between x = -2 and x = 3
y = 4x y axis between x = 0 and x = 4
y = √25 – x2 x axis
The above problems turn out to be easy since the solid becomes a standard geometric figure.

4. Volumes of Solids
What is the general method for finding the volume of any solid?
Assume we are given a solid that is not a familiar shape from Geometry.
Start by finding a “cross section” of the solid
Find the area of the cross section
“Cut” the solid into n cross sectional areas
Find the sum of volumes of each piece

5. Volumes of Solids So pretty easy process, right?....
If A(x) is the area of the cross section, the volume would be the area of the cross section times the height of each “piece.”
So the volume then is…

6. Volumes of Solids
Look familiar? Just as we did for area, we can rewrite the previous formula using integrals…
When finding the area of a region bound by the curve and x axis, we used rectangles.
What should we use for volume???

7. Volumes of Solids
Just as we used rectangles to find the area of a region, we will use cylinders (rectangles with a height) to find volumes of solids.
Guided Practice: page #2, 4

8. Disks vs. Washers
The disk method can be used when the cross section is a disk… A = πr2
If we want to revolve a region bounded by two curves, we will need to use the washer method.
Find the inner radius & outer radius, and then find the area…
A = π(outer radius)2 – π(inner radius)2

9. Practice and Homework Guided Practice Homework page 463 #6 – 16 (Even)
page #1 – 15 (Odd)