1. Solids of Revolution Shell Method
Section 7-3-A
Solids of Revolution
Shell Method

2. Cylindrical Shell Method
When an area between two curves is revolved about an axis a solid is created.
This solid is the sum of many, many concentric cylinders.
Volume = ∑circumference ∙ height ∙thickness

3. The Shell Method Consider a representative rectangle as shown
where w is the width
of the rectangle, h is the height of
the rectangle and rotation is
About the y-axis (vertical)

4. Shell Method: Horizontal Axis of Revolution
Slice parallel to the revolution axis
Vertical Axis of Revolution
Slice parallel to the revolution axis

5. Shell Method: V = 2prhw Horizontal Axis of Revolution
Integrate in terms y
Vertical Axis of Revolution
Integrate in terms x
Width
Radius
Height

6. Two Functions:
The volume of the solid of revolution of the region bounded by f(x) and g(x) is given by:

7. 1) Find the volume of the region bounded by
1) Find the volume of the region bounded by the x – axis, x = 1, and x = 4 revolve about the y – axis. 1 4

8. 2) Find the volume of the solid formed by revolving the region bounded by the graph of , the x – axis, and x = 9 about the y - axis. 9

9. 3) Find the volume of the solid formed by
3) Find the volume of the solid formed by revolving the region bounded by the graph of x = y3 and x = y from y = 0 to y = 1about the x - axis.
Need in terms of y 1

10. 4) Find the volume of the solid formed by
4) Find the volume of the solid formed by revolving the region bounded by the graph of x = y3 and x = y from y = 0 to y = 1 about the line y = -1. 1
y = -1

11. Homework Page 474 # 1-9, & 15-19, set up but do not integrate
# 23 find the volume using the shell method