Use cylindrical coordinates to evaluate the triple integral
$\iiint _ { E } y d V$
where $E$ is the solid that lies between the cylinders $x ^ { 2 } + y ^ { 2 } = 3$ and $x ^ { 2 } + y ^ { 2 } = 7$ above the $x y$-plane and below the plane $z = x + 4$.
Select the correct answer.
A) $8.57$
B) 0
C) $3.4$
D) $9.19$
E) $0.54$

Question

Quizplus · Accepted Answer

In cylindrical coordinates, the solid is defined by $3 \leq r^2 \leq 7$ and $0 \leq 	heta \leq 2\pi$ and $0 \leq z \leq r \cos 	heta + 4$. Thus, we have:
\begin{align*}
\iiint_E y\,dV &= \int_0^{2\pi} \int_{\sqrt{3}}^{\sqrt{7}} \int_0^{r \cos 	heta + 4} r\sin	heta \,dz\,dr\,d	heta \
&= \int_0^{2\pi} \int_{\sqrt{3}}^{\sqrt{7}} r\sin	heta(r\cos	heta + 4)\,dr\,d	heta \
&= \int_0^{2\pi} \left(\frac{r^3}{3}\cos	heta + 4r\sin	heta ight)_{\sqrt{3}}^{\sqrt{7}} \,d	heta \
&= \int_0^{2\pi} 0\,d	heta \
&= \boxed{0}.
\end{align*}

Use Cylindrical Coordinates to Evaluate the Triple Integral ∭EydV\iiint _ { E } y d V∭E​ydV

Use Cylindrical Coordinates to Evaluate the Triple Integral $\iiint _ { E } y d V$