Evaluate the triple integral triple integral_E (x + 1) dV, where E is the bounded by the...

Question:

Evaluate the triple integral

{eq}\displaystyle \iiint_E (x + 1)\ dV {/eq}, where E is the bounded by the paraboloid {eq}\displaystyle x = y^2 + z^2 {/eq} and the plane {eq}x = \pi^2 {/eq}.

Cylindrical Coordinates:

Given the situation above, we immediately want to use cylindrical coordinates. But we will need to change things a bit from the usual way we see them sice we need the axis of rotation to be {eq}x {/eq} instead of {eq}z {/eq}. We will be using the following:

{eq}x=x {/eq}

{eq}y = r \cos \theta {/eq}

{eq}z = r \sin \theta {/eq}

{eq}r^2 = y^2+z^2 {/eq}

{eq}\theta = \tan^{-1} \frac{z}{y} {/eq}

{eq}dV = r \ dx \ dr \ d\theta {/eq}