what is the volume generated by rotating the given region.

Question

My professor says the volume generated by rotating the region $\mathscr{R}_2$ about the line $OA$ is $5/\pi$ but I don't see how that could be the answer?

Answer 1

Hint: Assuming that the functions $f(x)$ and $g(x)$ are continuous and non-negative on the interval $[a, b]$ and $g(x) \le f(x)$, consider a region that is bounded by two curves $y=f(x)$ and $y=g(x)$, between $x=a$ and $x=b$.

The volume of the solid formed by revolving the region about the $x$-axis is

$$V=\pi\int_a^b\left(\left[ f(x)\right]^2 - \left[g(x)\right]^2 \right)dx.$$

So, we have:

$$V=\pi\int_0^1\left(1^2 - \sqrt{x}^2\right)dx=\pi\int_0^1\left(1 - x\right)dx=\boxed{\frac{\pi}{2}}.$$