Find the volume of the solid bounded by the cone y = sqrt(x^2 + z^2) and the paraboloid y = x^2 +...

Question:

Cylindrical Coordinates:

To solve the above integral, we will want to use cylindrical coordinates, but we will need to use a slightly modified version. We need the axis of rotation to be around the {eq}y {/eq}-axis, so exchange {eq}z {/eq} for {eq}y {/eq} in the usual cylindrical coordinates to get

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

{eq}y = y {/eq}

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

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

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

{eq}dV = r \; dy \; dr \; d\theta {/eq}

Answer and Explanation: 1