Evaluate the cylindrical coordinate integral.
-$\int _ { 0 } ^ { \pi / 2 } \int _ { 0 } ^ { \cos ^ { 2 } \theta } \int _ { 3 r ^ { 4 } } ^ { 9 r ^ { 4 } } z \sin \theta d z r d r d \theta$
A) $\frac { 39 } { 70 }$
B) $\frac { 1 } { 8 }$
C) $\frac { 6 } { 35 }$
D) $\frac { 51 } { 140 }$

Question

Quizplus · Accepted Answer

The integral evaluates to $\frac{6}{35}$ after performing the integration step-by-step: first with respect to $z$, then $r$, and finally $\theta$.

Evaluate the Cylindrical Coordinate Integral ∫0π/2∫0cos⁡2θ∫3r49r4zsin⁡θdzrdrdθ\int _ { 0 } ^ { \pi / 2 } \int _ { 0 } ^ { \cos ^ { 2 } \theta } \int _ { 3 r ^ { 4 } } ^ { 9 r ^ { 4 } } z \sin \theta d z r d r d \theta∫0π/2​∫0cos2θ​∫3r49r4​zsinθdzrdrdθ

Evaluate the Cylindrical Coordinate Integral $\int _ { 0 } ^ { \pi / 2 } \int _ { 0 } ^ { \cos ^ { 2 } \theta } \int _ { 3 r ^ { 4 } } ^ { 9 r ^ { 4 } } z \sin \theta d z r d r d \theta$