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Last modified: January 26, 2018
Volume Integrals
----------------
Disc Method
-----------
V = π∫[f(x)]2 dx
Basically, this represents a 360 ° rotation about the
x axis. The area swept out by the rotation is the area
of the circle with radius [f(x)]2.
Example: Sphere
Circle: x2 + y2 = r2
Semicircle: y = √(r2 - x2)
f(x)
^
| . .
| . .
| . .
|. .
------------------->x
-r 0 r
r
V = π∫y2 dx
-r
r
= π∫(r2 - x2) dx
-r
r
= [π(r2x - x3/3]
-r
= 4πr3/3
Shell Method
------------
Volume of concentric shell: dV = 4πr2dr
Therefore,
r
V = ∫4πr2dr = 4πr3/3
0