Shell integration

Consider the ring-cake shape created by rotating the region bounded by the line x = 0, the line ${\displaystyle x={\sqrt {\pi /2}}}$, and the curve ${\displaystyle y=\sin(x^{2})}$ in the x-y-plane, around the y-axis into the z-dimension.^[1] We can divide this shape into circular city-wall shells. The length of each shell is its circumference, which is ${\displaystyle 2\pi \cdot radius}$: ${\displaystyle 2\pi x}$ in this context. The width of the shell is dx, so in two dimensions the area is ${\displaystyle 2\pi xdx}$. The height of the shell is ${\displaystyle y=f(x)=\sin(x^{2})}$, so the volume of each shell is ${\displaystyle 2\pi x\sin(x^{2})dx}$. Adding up all the shells as x changes, we come out with

${\displaystyle volume=\int _{0}^{\sqrt {\pi /2}}2\pi x\sin(x^{2})dx=\int _{0}^{\pi /2}\pi \sin(u)du={\Big |}_{0}^{\pi /2}-\pi \cos(u)=\pi (0--1)=\pi ,}$

where this integral has been solved by the method of substitution setting ${\displaystyle u=x^{2}}$ so ${\displaystyle du=2xdx}$ and the bounds change from 0 and ${\displaystyle {\sqrt {\pi /2}}}$ to 0 and ${\displaystyle \pi /2}$.

Keep the old example too,e ven tho it is not as well written. Two examples are good to have. Comment on the Talk page on what I have done, and its deficiencies.