Convert the integral below from rectangular coordinates to both cylindrical and spherical coordinates, and evaluate the simpler iterated integral. $ \int_{0}^{9} \int_{0}^{\sqrt{81-x^{2}}} \int_{0}^{\sqrt{81-x^{2}-y^{2}}} \sqrt{x^{2}+y^{2}+z^{2}} d z d y d x $
A) $\frac { 729 } { 8 } \pi$
B) $\frac { 6561 } { 4 } \pi$
C) $6561 \pi$
D) $\frac { 6561 } { 2 } \pi$
E) $\frac { 6561 } { 8 } \pi$

Question

Quizplus · Accepted Answer

The given integral represents a volume integral over a sphere of radius 9. In spherical coordinates, the integral simplifies to $\int_{0}^{2\pi} \int_{0}^{\pi} \int_{0}^{9} \rho^3 \sin(\phi) d\rho d\phi d\theta$, which evaluates to $\frac{6561}{8} \pi$.

Convert the Integral Below from Rectangular Coordinates to Both Cylindrical ∫09∫081−x2∫081−x2−y2x2+y2+z2dzdydx \int_{0}^{9} \int_{0}^{\sqrt{81-x^{2}}} \int_{0}^{\sqrt{81-x^{2}-y^{2}}} \sqrt{x^{2}+y^{2}+z^{2}} d z d y d x ∫09​∫081−x2​​∫081−x2−y2​​x2+y2+z2​dzdydx

Convert the Integral Below from Rectangular Coordinates to Both Cylindrical $\int_{0}^{9} \int_{0}^{\sqrt{81-x^{2}}} \int_{0}^{\sqrt{81-x^{2}-y^{2}}} \sqrt{x^{2}+y^{2}+z^{2}} d z d y d x$