Evaluate using spherical coordinates: triple integral over D of (x^2 + y^2 + z^2)^(3/2) dxdydz...

Question:

Evaluate using spherical coordinates: {eq}\iiint_{D} (x^2 + y^2 + z^2)^{\frac{3}{2}} \, \mathrm{d}x \, \mathrm{d}y \, \mathrm{d}z {/eq} where {eq}D = \left \{ (x, y, z): x^2 + y^2 + z^2 \leq 9 \right \} {/eq}.

Answer and Explanation:

For the integral

{eq}\iiint_{D} (x^2 + y^2 + z^2)^{\frac{3}{2}} \, \mathrm{d}x \, \mathrm{d}y \, \mathrm{d}z {/eq} where {eq}D = \left \{ (x, y, z): x^2 + y^2 + z^2 \leq 9 \right \} {/eq}, we can parametrize {eq}D {/eq} by {eq}D = \{(\rho,\theta,\varphi): 0\le \varphi\le 3,\, 0\le\varphi\le2\pi,\, 0\le\theta\le2\pi \} {/eq}

Hence,

{eq}\begin{align*} \iiint_{D} (x^2 + y^2 + z^2)^{\frac{3}{2}} \, \mathrm{d}x \, \mathrm{d}y \, \mathrm{d}z & = \int_0^3\int_0^{2\pi}\int_0^{2\pi} \rho^5\sin\varphi d\varphi d\theta d\rho\\ & = \int_0^3\int_0^{2\pi}\int_0^{2\pi} \rho^5\sin\varphi d\varphi d\theta d\rho = 0. \end{align*} {/eq}


Learn more about this topic:

Cylindrical & Spherical Coordinates: Definition, Equations & Examples

from GRE Math: Study Guide & Test Prep

Chapter 13 / Lesson 10
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