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Find the volume of the solid bound by {eq}z = 4 - (x^2 + y^2) {/eq} and z = 0.

Question:

Find the volume of the solid bound by {eq}z = 4 - (x^2 + y^2) {/eq} and z = 0.

Cylindrical Coordinates:

Recall that we can find the volume of region by setting up and evaluating a triple integral. Since our region possesses axial symmetry, we will find it convenient to use cylindrical coordinates. Recall

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

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

{eq}z = z {/eq}

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

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

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

Answer and Explanation: 1

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We have {eq}0 \leq z \leq 4-r^2 {/eq}. We can bound {eq}r {/eq} at the intersection of the surfaces.

{eq}\begin{align*} 4-r^2 &= 0 \\ r^2 &= 4...

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Cylindrical & Spherical Coordinates: Definition, Equations & Examples

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Chapter 13 / Lesson 10
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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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