Find volume of region with cylinders and planes?

Question

I need help with this integral as well:

the region bounded by the cylinders r=2 r=3 and the planes z=8 and z=10

Answer 1

This problem is easiest in cylindical coordinates.

The region in the xy plane is between the concentric circles given by r=2 and r=3.  So the limits on r will be r goes 2 to 3.

Θ will go from 0 to 2pi

z will go from z=8 to 10.

The volume will be given by the triple integral of 1 dV, but recall that changing to cylindrical introduces the factor r:

∫∫∫ 1 dV = ∫∫∫ rdrdΘdz  which works out to 10 pi.

Let me know if you need more detail.