Volume of $y = 6\sqrt{\sin(x)}$ rotated around $y$-axis using triple integrals

Question

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.

Answer 1

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.