MHB Huzafa's question at Yahoo Answers regarding a solid of revolution

[MarkFL]

Here is the question:

Find volume of solid obtained by revolving around y-axis the plane area btw the graph y=1-x^2 and the x-axis?

I have posted a link there to this topic so the OP can see my work.

 

[MarkFL]

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:

$$V=2\pi\int_0^2 x-x^3\,dx=\frac{\pi}{2}\left[2x^2-x^4 \right]_0^1=\frac{\pi}{2}$$