Integrate x^{2}+y^{2}+z^{2} over the cylinder x^{2}+y^{2}\leq 2,2\leq z\leq 3.

Question:

Integrate {eq}x^{2}+y^{2}+z^{2} {/eq} over the cylinder {eq}x^{2}+y^{2}\leq 2,2\leq z\leq 3 {/eq}.

Cylindrical Coordinates:

We use cylindrical coordinates for integration.

In cylindrical coordinates {eq}x^2+y^2=r^2. {/eq}

The projection of the region of integration on the plane should be treated as a region in polar coordinates.

In cylindrical coordinates, the volume element is {eq}dz\, r\, dr\, d\theta. {/eq}

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We are asked to integrate a three-variable function over a region in space. The region of integration contains a cylinder. So it is better to use... Cylindrical & Spherical Coordinates: Definition, Equations & Examples

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Chapter 13 / Lesson 10
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Cylindrical coordinates and spherical coordinates are similar to the cartesian coordinate system but applied to specifying points in a three-dimensional plane. Learn how polar coordinates apply to these calculations and their use in celestial maps.