Let $R$ be the solid in $\mathbb{R}^{3}$ bounded by the surfaces $x=0, y=0, z=0$ and $x^{2}+y^{2}+z^{2}=1$. Evaluate
$$
\iiint_{R} 3\left(x^{3}+x y^{2}+x z^{2}\right) d V
$$
$\frac{\pi}{4}$
$\frac{\pi}{32}$
$\frac{\pi}{8}$
$\frac{\pi}{16}$

Question

Let $R$ be the solid in $\mathbb{R}^{3}$ bounded by the surfaces $x=0, y=0, z=0$ and $x^{2}+y^{2}+z^{2}=1$. Evaluate
$$
\iiint_{R} 3\left(x^{3}+x y^{2}+x z^{2}\right) d V
$$
$\frac{\pi}{4}$
$\frac{\pi}{32}$
$\frac{\pi}{8}$
$\frac{\pi}{16}$

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ Convert the integral to spherical coordinates because the region of integration is a sphere in the first octant. ‖ ‖ step 2 ⋮ The spherical coordinates are given by $ x = \rho \sin(\phi) \cos(\theta) $, $ y = \rho \sin(\phi) \sin(\theta) $, and $ z = \rho \cos(\phi) $. ‖ ‖ step 3 ⋮ The Jacobian determinant for the transformation to spherical coordinates is $ \rho^2 \sin(\phi) $. ‖ ‖ step 4 ⋮ Set up the limits of integration for $ \rho $ from 0 to 1, $ \phi $ from 0 to $ \frac{\pi}{2} $, and $ \theta $ from 0 to $ \frac{\pi}{2} $ to cover the first octant of the sphere. ‖ ‖ step 5 ⋮ The integral becomes $ \int_{0}^{\frac{\pi}{2}} \int_{0}^{\frac{\pi}{2}} \int_{0}^{1} 3(\rho^3 \sin^3(\phi) \cos^3(\theta) + \rho^3 \sin^3(\phi) \cos(\theta) \sin^2(\theta) + \rho^3 \sin^3(\phi) \cos(\theta) \cos^2(\phi)) \rho^2 \sin(\phi) d\rho d\theta d\phi $. ‖ ‖ step 6 ⋮ Integrate with respect to $ \rho $, then $ \theta $, and finally $ \phi $. ‖ ‖ step 7 ⋮ After performing the integration, the result is $ \frac{\pi}{4} $. ‖ ∻Answer∻ ⚹ $ \frac{\pi}{4} $ ⚹ ∻Key Concept∻ ⚹Spherical Coordinates in Integration⚹ ∻Explanation∻ ⚹The triple integral over a spherical region is often easier to evaluate using spherical coordinates, which simplifies the integration process when the function and limits are symmetrical with respect to the origin.⚹◊What is the volume of the solid region $R$ bounded by the surfaces $x=0, y=0, z=0$, and $x^{2}+y^{2}+z^{2}=1$ in $\mathbb{R}^{3}$?⍭ Generate me a similar question◊

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