1. Volumes of Revolution: Disk and Washer Method
AP Calculus

3. Volume of Revolution: Method
Lengths of Segments:
In revolving solids about a line, the lengths of several segments are needed for the radii of disks, washers, and for the heights of cylinders.
A). DISKS AND WASHERS
1) Shade the region in the first quadrant (to be rotated)
2) Indicate the line the region is to be revolved about.
3) Sketch the solid when the region is rotated about the indicated line.
4) Draw the representative radii, its disk or washer and give their lengths.
<<REM: Length must be positive! Top – Bottom or Right – Left >>
Ro = outer radius
ri = inner radius

4. Disk Method The Formula: The formula is based on the
Rotate the region bounded by f(x) = 4 – x2 in the first quadrant about the y - axis
The region is _______________ _______ the axis of rotation.
The Formula: The formula is based on the
_____________________________________________

5. Washer Method The Formula: The formula is based on
Rotate the region bounded by f(x) = x2, x = 2 , and y = 0 about the y - axis
The region is _______________ __________ the axis of rotation.
The Formula: The formula is based on
_____________________________________________

6. Disk Method
Rotate the region bounded by f(x) = 2x – 2 , x = 4 , and y = 0 about the line x = 4
The region is _______________ _______ the axis of rotation.

7. Washer Method
Rotate the region bounded by f(x) = -2x , x = 2 , and y = 0 about the y - axis
The region is _______________ __________ the axis of rotation.

8. Example 1:
The region in the first quadrant is bounded by Rotated about:
, the x-axis, and the y-axis a) The x-axis
b) The y-axis
c) x = 3
d) y = 4

9. Example 2: The region is bounded by: Rotated about:
f(x) = x and g(x) = x2 a) the x-axis
in the first quadrant b) the y-axis
c) x = 2
d) y = 2