Calculating Volume of Solid Using Triple Integral

[physics&math]

1. Use a triple integral to find the volume of the given solid.

The solid enclosed by the cylinder x^2 + z^2 = 4 and the planes y = -1 and y + z = 4


This looked like a cylindrical coordinate system to me, except for the fact that it is not cylindrical around the z-axis but the y-axis. I tried to fix this problem by "rotating" my coordinate axes so that my old z-axis would be my new x-axis, x would become y, and y would become z. I'm not sure if this is a valid approach or not.

 

[hilbert2]

It makes no difference mathematically how we name the coordinate axes. The coordinate system $(r,\theta,y)$, with
$x=rcos\theta$
$z=rsin\theta$,
is as valid a cylindrical coordinate system as the usual $(r,\theta,z)$.

Now the volume element is $dxdydz=rdrd\theta dy$. Note that because of the plane $y+z=4$ all of the integration limits in the volume calculation are not constants, one of them is a function of $r$ and $\theta$.

 

[brmath]

Hilbert1 is absolutely correct. However, if you lack his sophistication, you can just rewrite the problem switching the y and z. They are just symbols, right?

 

[physics&math]

Thank you both so much!