Hello huzafa,
The first thing I would do is draw a diagram of the region to be revolved. We need only concern ourselves with either the quadrant I area or the quadrant II area because we are revolving an even function about the $y$-axis. I will choose to plot the quadrant I area:
View attachment 974
Using the disk method, we observe that the volume of an arbitrary disk is:
$$dV=\pi r^2\,dy$$
where:
$$r=x\,\therefore\,r^2=x^2=1-y$$
and so we have:
$$dV=\pi(1-y)\,dy$$
Summing the disks by integration, we have:
$$V=\pi\int_0^1 1-y\,dy=\pi\int_0^1 u\,du=\frac{\pi}{2}\left[u^2 \right]_0^1=\frac{\pi}{2}$$
Using the shell method, we observe that the volume of an arbitrary shell is:
$$dV=2\pi rh\,dx$$
where:
$$r=x$$
$$h=y=1-x^2$$
and so we have:
$$dV=2\pi\left(x-x^3 \right)\,dx$$
Summing the shells by integration, we find:
$$V=2\pi\int_0^2 x-x^3\,dx=\frac{\pi}{2}\left[2x^2-x^4 \right]_0^1=\frac{\pi}{2}$$
