Use the Shell Method to find the volume by revolving around the x-axis.

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The functions are $x=\frac{y^2}{2}$, $x=2$, and $y=2$. I graphed it and it looks like the intersection points are $(2,2)$ and $(0,2)$.

But I don't know how to set up the integral.

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1 Answer

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For the shell method, finding the volume when rotating about the $x$-axis, we have the general formula:

$$\int_a^b 2\pi\,r_y\,h_y\,dy$$

In this case, we have $a = 0, b= 2$, $r_y = y$ ("radius"), and height $h_y$ given by the curve $h_y = \frac {y^2}{2}$.

That gives us $$\int_0^2 2\pi\, y\,\left(\frac {y^2}{2}\right)\,dy \quad= \quad\pi\int_0^2 y^3\,dy\quad = \quad \frac{\pi}4 y^4\,\Big]_0^2$$

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