I'm trying to find the volume of the solid of revolution formed by revolving the curve whose equation is given around the X-axis between the points indicated. I would like your feedback on my solution.
$$\pi\int^{1}_{0} \left(\prod_{k=0}^{n}\frac{x^k}{k!}\right)^2$$
My working out:
$$\pi\ \frac{1}{\prod_{k=0}^{n} k!}\int_{0}^{1} \left(x^{\sum^{n}_{k=0}}k\right)^2 \space dx= $$$$\pi\ \frac{1}{\prod_{k=0}^{n} k!}\int_{0}^{1} (x^{n(n+2)}) \space dx = \pi\ \left[\frac{x^{n(n+2)+1}}{(n(n+2)+1)\prod_{k=0}^{n} k!}\right]^{1}_{0}=\left[\frac{\pi}{(n(n+2)+1)\prod_{k=0}^{n} k!}\right]$$
This approach is new to me, as I've just seen this approach in a previous answer to my question here, so I'd thought that I would practice with this equation. Please let me know of my effort.