Question

What will be the volume of the cylinder: `x^2 + y^2 = 4` and the planes `y+z=4` and `z=0`?

Answer 1

`16 pi`

Explanation:

Cylindrical Volume:

`V = int_V r \ dr \ dz \ d theta`

`= int_(theta = 0)^(2 pi) int_(r=0)^(2) int_(z=0)^(4 - r sin theta) \ r \ dz \ dr \ d theta`

`= int_(theta = 0)^(2 pi) int_(r=0)^(2) \ (rz)_(z=0)^(4 - r sin theta) \ dr \ \ d theta`

`= int_(theta = 0)^(2 pi) int_(r=0)^(2) \ 4 r - r^2 sin theta \ dr \ \ d theta`

`= int_(theta = 0)^(2 pi) ( \ 2r^2 - r^3/3 sin theta )_(r=0)^(2) \ \ d theta`

`= int_(theta = 0)^(2 pi) \ 8 - 8/3 sin theta \ \ d theta`

`= ( \ 8theta + 8/3 cos theta \ )_(theta = 0)^(2 pi)`

`= 16pi`

This is same as:

• `pi (2^2) * 2 + pi (2^2) * (6 - 2)/2 = 16 pi`

...which you get if you just splice and dice the cylinder using symmetry