Set up and compute a double integral in polar coordinates to find the volume of a cylinder of...

Question:

Set up and compute a double integral in polar coordinates to find the volume of a cylinder of height {eq}1{/eq} over the region {eq}x^2 + y^2 \le 1. {/eq}

Volume of the Region:

The double integrals formula for the volume of the region in polar coordinates is {eq}V=\int \int zrdrd\theta {/eq} where {eq}z {/eq} is the length or height and {eq}\int \int rdrd\theta {/eq} represents the area.

Answer and Explanation:

Based on the given equations the limits are the following,

{eq}0\leq \theta \leq 2\pi ,\:0\leq r\leq 1 {/eq}

The volume of the region is,

{eq}V=\int_{0}^{2\pi }\int_{0}^{1}zrdrd\theta {/eq}

{eq}V=\int_{0}^{2\pi }\int_{0}^{1}(1)rdrd\theta {/eq}

Integrate with respect to {eq}r {/eq}

{eq}V=\int_{0}^{2\pi }\left [ \frac{1}{2}r^{2} \right ]^{1}_{0}d\theta {/eq}

{eq}V=\int_{0}^{2\pi }\frac{1}{2}d\theta {/eq}

Integrate with respect to {eq}\theta {/eq}

{eq}V=\frac{1}{2}\left [ \theta \right ]^{2\pi }_{0} {/eq}

{eq}V=\pi {/eq}


Learn more about this topic:

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Volumes of Shapes: Definition & Examples

from GMAT Prep: Tutoring Solution

Chapter 11 / Lesson 9
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