use a triple integral to find the volume of the given solid
the solid enclosed by the cylinder x^2+y^2=9 and the planes y+z=5 and z=1
i attempted to draw a picture..
and i ended up with the following integral:
(i don't know how to write math symbols on here)
int from 0 to 2pi int from 0 to 3 int from 1 to y-5 of dzrdrdtheta
i'm positive that's wrong.
since it's a cylinder, i figured polar seemed like a good idea.. and i checked the answer in the back and it's 36pi. the pi leads me to think i should use polar.
so my first limits are 0 and 2pi followed by 0 and 3, since 3 is the radius of the cylinder.
it is after that that i get very confused.
i think i should be integrating from 1 to y=5 because those are the two planes when set equal to z...
but i'm not sure.
and i never know what to actually integrate, so i just had dzrdrdtheta...
the answer is 36pi, i just don't know how to get there.
so i am sure there are still a lot of stupid arithmetic errors in my work that are causing this but i can't find them:
int 0 to 2pi int 0 to 3 (5r - r^2sintheta)drdtheta
= int 0 to 2 pi (5r^2/2 - (r^3)sintheta/3 from 0 to 3 dtheta
= int 0 to 2pi (45/2 - 9sintheta) dtheta
=45theta/2 + 9costheta from 0 to 2pi
=45pi + 9cos2pi - 9cos0
=45pi + 9 - 9?