Evaluate the cylindrical coordinate integral.
-$\int _ { 0 } ^ { 9 \pi } \int _ { 0 } ^ { 6 } \int _ { 0 } ^ { r } \left( 3 r ^ { 2 } + 4 z ^ { 2 } \right) d z r d r d \theta$
A) $\frac { 303,264 } { 5 } \pi$
B) $75,816 \pi$
C) $\frac { 454,896 } { 5 } \pi$
D) $101,088 \pi$

Question

Quizplus · Accepted Answer

To solve the given cylindrical coordinate integral, we first integrate with respect to $z$, then $r$, and finally $\theta$. The integral is:$$\int _ { 0 } ^ { 9 \pi } \int _ { 0 } ^ { 6 } \int _ { 0 } ^ { r } \left( 3 r ^ { 2 } + 4 z ^ { 2 } \right) d z r d r d \theta$$First, integrate with respect to $z$:$$\int _ { 0 } ^ { r } \left( 3 r ^ { 2 } + 4 z ^ { 2 } \right) d z = 3 r ^ { 2 } z + \frac{4}{3} z ^ { 3 } \Big|_0^r = 3 r ^ { 3 } + \frac{4}{3} r ^ { 3 } = 3 r ^ { 3 } + \frac{4 r ^ { 3 }}{3} = \frac{9 r ^ { 3 } + 4 r ^ { 3 }}{3} = \frac{13 r ^ { 3 }}{3}$$Then, integrate with respect to $r$:$$\int _ { 0 } ^ { 6 } \frac{13 r ^ { 3 }}{3} r d r = \frac{13}{3} \int _ { 0 } ^ { 6 } r ^ { 4 } d r = \frac{13}{3} \cdot \frac{r ^ { 5 }}{5} \Big|_0^6 = \frac{13}{3} \cdot \frac{6 ^ { 5 }}{5} = \frac{13 \cdot 7776}{15} = \frac{101088}{15}$$Finally, integrate with respect to $\theta$:$$\int _ { 0 } ^ { 9 \pi } \frac{101088}{15} d \theta = \frac{101088}{15} \cdot 9 \pi = \frac{303264}{5} \pi$$Therefore, the correct answer is $\frac { 303,264 } { 5 } \pi$, which is choice A.

Evaluate the Cylindrical Coordinate Integral ∫09π∫06∫0r(3r2+4z2)dzrdrdθ\int _ { 0 } ^ { 9 \pi } \int _ { 0 } ^ { 6 } \int _ { 0 } ^ { r } \left( 3 r ^ { 2 } + 4 z ^ { 2 } \right) d z r d r d \theta∫09π​∫06​∫0r​(3r2+4z2)dzrdrdθ

Evaluate the Cylindrical Coordinate Integral $\int _ { 0 } ^ { 9 \pi } \int _ { 0 } ^ { 6 } \int _ { 0 } ^ { r } \left( 3 r ^ { 2 } + 4 z ^ { 2 } \right) d z r d r d \theta$