How to find A for solids of revolution:

if cross-section is a disk

if cross-section is a washer22

2

)()(

)(

radiusinnerradiusouterA

radiusA

b

a

b

a

dyyAVordxxAV )()(Volume of solids by circular cylinders:

Not solids of revolution:

Volume of solids by cylindrical shells:

hrrhrrrr

hrhrVVV

2)(2

2 12212

12

212

Region bounded by y=f(x)0, y=0, x=a, x=b. Rotation about y-axis.

dxxfxVb

a

)(2