1. Volumes by
Cylindrical
Shells
Rita Korsunsky

2. Formula for the volumes by Shells method
For volume of a solid, add up all the volumes of each shell:
radius dx
f(x) = height

3. How to draw the radius and the height
1) Take a point on the curve
2) Draw the radius perpendicular to the axis of rotation
3) Drop a height from the point parallel to the axis of rotation

4. Example#1 dx
radius
2x – x2 x
height 2

5. Example#2 y dx
radius
y = x2
height
3 - x 3
y = x + 2 x

6. Example#3 2
height dy y
radius
2y3 – y4

7. Example#4 R r
Find points of intersection:
Using Washers Method:

8. Example#4
Or using Shell’s Method:

9. The End