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


Find the volume of the solid bounded by the cone {eq}y = \sqrt{x^2 + z^2} {/eq} and the paraboloid {eq}y = x^2 + z^2 {/eq}.

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

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In cylindrical coordinates, our region is {eq}r^2 \leq y \leq r {/eq} on the interval {eq}r \in [0,1] {/eq}. Since, we are rotating all the way...

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Cylindrical & Spherical Coordinates: Definition, Equations & Examples


Chapter 13 / Lesson 10

Learn how to convert between Cartesian, cylindrical and spherical coordinates. Discover the utility of representing points in cylindrical and spherical coordinates.

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