Volume of $y = 6\sqrt{\sin(x)}$ rotated around $y$-axis using triple integrals The problem is to find the volume of $y = 6\cdot \sqrt{\sin (x)}$ rotated around the $y$-axis when $0 \leq y \leq 6$.
I know this can be done by the sv-calc method of volumes of revolution but I wanted to see if a problem like this can be done by triple integrals. I tried it a few times myself but could not seem to get the limits of the integrals set up correctly.
 A: Maybe I miss the point of your question, but I think that the rotation of a function $y=f(x)$ around the $y$-axis using the formula
\begin{align*}
V=\pi\int_a^bx^2(y) dy\tag{1}
\end{align*}
and your Ansatz in the comment using a triple integral is essentially the same.

Since for 
\begin{align*}
y=6\sqrt{\sin (x)}\qquad\longleftrightarrow\qquad x=\arcsin\left(\frac{y^2}{36}\right)
\end{align*}
  We observe according to your comment
  \begin{align*}
&\int_0^6\int_0^{\arcsin(\frac{y^2}{36})}\int_0^{2\pi}r d\Theta dr dy\\
&\qquad=\int_0^6\int_0^{\arcsin(\frac{y^2}{36})}\left. r\cdot\Theta\right|_0^{2\pi} dr dy\\
&\qquad=2\pi\int_0^6\int_0^{\arcsin(\frac{y^2}{36})}r dr dy\\
&\qquad=2\pi\int_0^6\left.\left(\frac{1}{2}r^2\right)\right|_0^{\arcsin(\frac{y^2}{36})}dy\\
&\qquad=\pi\int_0^6\arcsin^2\left(\frac{y^2}{36}\right)dy\tag{2}\\
\end{align*}
and (2) corresponds to the volume formula (1).

I think the difficulty lies in solving the integral (2) which seems to allow no simple closed representation. Wolfram alpha provides following solution:

\begin{align*}
&\pi\int\arcsin^2\left(\frac{y^2}{36}\right)dy\\
&\qquad=\frac{\pi^2y^5}{5184\sqrt{2}\Gamma\left(\frac{7}{4}\right)\Gamma\left(\frac{9}{4}\right)}
_{3}F_{2}\left(1,\frac{5}{4},\frac{5}{4};\frac{7}{4},\frac{9}{4};\left(\frac{y}{6}\right)^4\right)\\
&\qquad\qquad-\frac{2\pi y}{3} _2F_1\left(1,\frac{5}{4};\frac{7}{4};\left(\frac{y}{6}\right)^4\right)
\arcsin\left(\frac{y^2}{36}\right)\sin\left(2\arcsin\left(\frac{y^2}{36}\right)\right)\\
&\qquad\qquad+\pi y\arcsin^2\left(\frac{y^2}{36}\right)+C
\end{align*}

Note: You could perform a plausibility check, take a simpler integrand and you'll be able to calculate the volume in both ways.
