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

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