Using spherical coordinates, the triple integral $\iiint$_E $e ^ { \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) ^ { \frac { 1 } { 2 } } } d V$ where E is the solid $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \leq 1$ is A) $\frac { 2 \pi ( e - 1 ) } { 3 }$ B) $\frac { 2 \pi ( e + 1 ) } { 3 }$ C) $\frac { 4 \pi ( e - 1 ) } { 3 }$ D) $\frac { 4 \pi ( e + 1 ) } { 3 }$ E) $\frac { 2 \pi ( e - 1 ) } { 3 }$.

Question

Quizplus · Accepted Answer

The given integral in spherical coordinates for the solid $E$ (a sphere of radius 1) is $\iiint e^{\rho} \rho^2 \sin(\phi) d\rho d\phi d\theta$, where $\rho$ is the radius, $\phi$ is the angle with the positive z-axis, and $\theta$ is the angle in the xy-plane. The limits for $\rho$ are from 0 to 1, for $\phi$ from 0 to $\pi$, and for $\theta$ from 0 to $2\pi$. Integrating gives $\frac{4\pi(e-1)}{3}$.

Using Spherical Coordinates, the Triple Integral ∭\iiint∭ E e(x2+y2+z2)12dVe ^ { \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) ^ { \frac { 1 } { 2 } } } d Ve(x2+y2+z2)21​dV

Using Spherical Coordinates, the Triple Integral $\iiint$ _E $e ^ { \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) ^ { \frac { 1 } { 2 } } } d V$