1. 5053 -Volume by Shells
AP Calculus

2. Volume by Shells
V = l * w * h
( 2  r * x * f (x)

3. Formula:
Region may be either adjacent to or separate from the axis of rotation.
Representative Rectangle is Parallel to the axis of rotation.
“Parallel therefore shell.”

4. r = the distance from the axis of rotation.
Formula:
Then formula is built on the surface area of a cylinder ….
times …. its thickness.
r = the distance from the axis of rotation.
h = difference between the curves ( n )

5. r and h
Perpendicular to x - axis

6. r and h
Perpendicular to y - axis

7. Volume by Shells V = l * w * h ( 2  r * x * f (x) a) ||  shell b)
c) r = ( x - 0)
d) h = (f (x) - 0)

8. Example:
Revolve the region about the y – axis. 1

9. Example:
Rotated about x = 2

10. Example:
Rotated about x = 3

11. Example:
Rotated about y = 8