Find volume from rotating region bounded by $y=e^x$, $y=\cos(πx^2/2)$ and $x=1$ about $y$-axis

Question

Find the volume of the solid generated by rotating the region bounded by the curves $$y=e^x,\>\>\>\>\>y=\cos(πx^2/2),\>\>\>\>\>x=1$$

about the $y$-axis.

Answer 1

Use the shell method to integrate over $x$ as follows,

$$\int\limits_0^1 2\pi x\left(e^x-\cos\frac{\pi x^2}2\right)dx = 2(\pi-1)$$