Use cylindrical coordinates to find volume

[Carmen12]

Homework Statement

Use cylindrical coordinates to find volume...

Homework Equations

Inside: x²+y²+z²=16
Outside: z=sqrt(x²+y²)

The Attempt at a Solution

Cylindrical coordinates have always been a problem for me, so I initially tried to put them into spherical and then convert them over, but that went badly pretty quickly. Then basically integrated from 0 to 4 the inside minus the outside function but that didn't go well either... and obviously so.

So.. should I just try to convert from Cartesian to cylindrical and then integrate, but I have trouble with the bounds. :(

 

[Pepo]

In cylindrical coordinates you have 3 dimensions, r,Ø,z

The conversion from cartesian to cylindrical is

x=rCosØ
y=rSinØ
r=sqrt(x^2+y^2)

So you equation will be:

Outside:
r^2*cos^2(Ø)+r^2*sin^2(Ø)+z^2=16
=> r^2*(cos^2(Ø)+sin^2(Ø))+z^2=16 ; cos^2(Ø)+sin^2(Ø)=1 is a trigonometric identity
then

r^2+z^2=16
inside:
Z=r