Question

triple integral (cylindrical) 0 to 2pi, 0 to 3, 0 to Z/3 of r^3 dr dz dtheta; I have did this several times but I cannot get the answer of 3pi/10 some one catch my mistake?

Answer 1

can you show your work

Answer 2

\[\int\limits _{0}^{2\pi} \int\limits_{0}^{3} \int\limits_{0}^{\frac{z}{3}} r^3drdzd \theta\]
\[=\int\limits _{0}^{2\pi} \int\limits_{0}^{3} \left[\frac{r^4}{4} \right]_{0}^{\frac{z}{3}} dzd \theta\]
\[= \int\limits _{0}^{2\pi} \int\limits_{0}^{3} \frac{z^4}{324} dzd \theta\]
\[=\int\limits _{0}^{2\pi} \left[\frac{1}{324} \frac{z^5}{5} \right]_{0}^{3} d \theta\]
\[=\int\limits\limits _{0}^{2\pi} \frac{243}{1620}d \theta \rightarrow \int\limits\limits _{0}^{2\pi} \frac{3}{20}d \theta\]
\[=\left[ \frac{3}{20}\theta \right]_{0}^{2\pi}= \frac{6\pi}{20}= \frac{3\pi}{10}\]