Question: Evaluate $\iiint xyz \,dx\,dy\,dz$, over the positive octant of the sphere $x^2+y^2+z^2=a^2$ .
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Subject : Applied Mathematics 2

Topic : Triple integration and Applications of Multiple integrals

Difficulty : Medium

am2(82) • 503 views
Transforming to spherical coordinate by putting $x = r \hspace{0.1cm}sin \theta cos \theta , y=r \hspace{0.1cm}sin \theta \hspace{0.1cm}sin \phi, z=r \hspace{0.1cm}cos \theta$ and $dx\hspace{0.1cm}dy\hspace{0.1cm}dz = r^2sin \theta \hspace{0.1cm}dr \hspace{0.1cm}d \theta \hspace{0.1cm}d \phi$
$I = \int^{\pi/2}_{\phi=0}\int^{\pi/2}_{\theta=0}\int^a_{r=0} r^5\hspace{0.1cm}sin^3 \theta\hspace{0.1cm}cos \theta \hspace{0.1cm}sin \phi\hspace{0.1cm} cos \phi \hspace{0.1cm} dr \hspace{0.1cm} d\theta \hspace{0.1cm} d\phi\\ \hspace{0.1cm}=\int_0^{\pi/2}sin \phi \hspace{0.1cm}cos \phi \hspace{0.1cm}d\phi \int_0^{\pi/2} sin \theta \hspace{0.1cm}cos \theta \hspace{0.1cm} d \theta \int_0^ar^5\hspace{0.1cm}dr\\ \hspace{0.1cm}=\big[\frac{sin^2\theta}{2}\big]_0^{\pi/2}\big[\frac{sin^4\theta}{4}\big]_0^{\pi/2}\big[\frac{r^6}{6}\big]_0^a\\ \hspace{0.1cm}=\frac{1}{2}.\frac{1}{4}.\frac{a^6}{6} = \frac{a^6}{48}$