Answer
$\pi$
Work Step by Step
Our aim is to compute the volume of the given curve when it is revolved around $y$-axis by using the shell method.
Shell method for computing the volume of the curve: Let us consider two functions $f(x)$ and $g(x)$ (both are continuous functions) with $f(x) \geq g(x)$ on the interval $[m, n]$ and $R$ defines the region which revolves about the y-axis and is known as the region bounded by the curves $y=f(x)$ and $y=g(x)$ between the lines $x=m$ and $x=n$. Then, the volume of the solid can be expressed as:
$Volume, V=\int_m^n 2 \pi x [f(x)-g(x)] \ dx$
Here, $2\pi x=\text{Shell Circumference}$ and $[f(x)-g(x)] =\text {Height of the shell}$
We are given that $y=\cos (x^2); y=0$ on $[0, \sqrt {\dfrac{\pi}{2}}]$
Thus, $V=\int_m^n 2 \pi x [f(x)-g(x)] \ dx=2 \pi \int_0^{\sqrt {\frac{\pi}{2}}} x[\cos (x^2)] \ dx$
Suppose that $a=x^2 \implies da=2x dx$
Now, $V= \pi \int_0^{\pi/2} [\cos (a)] \ da$
or, $= \pi [ \sin a]_0^{\pi/2}$
or, $= \pi [\sin \dfrac{\pi}{2}-\sin (0)] $
or, $=\pi$
