Question

Find the volume of the region R enclosed by the curves y=x and y=x^2. And the rotated about the x-axis

Find the volume of the region R enclosed by the curves y=3sqrt(sin2x) and y=0. And the rotated about the x-axis

Answer 1

For both of these question I was confused as to what to which method to use: A=pi*r^2 or A=pi*r^2-pi*r^2

I also was confused for what bounds to use for the second problem?

Thanks

Answer 2

When the bounded region of interest is contiguous with the axis of revolution for the entire interval, you would use the disk method (the \(\pi r^2\) form). If there is a gap or space between the region and the axis, you would use the washer method (the \(\pi({r_1}^2-{r_2}^2)\) form.)

Here's the distinction: In the sketches below, the axis of revolution is the x-axis. The case on the left is the contiguous case, and the one on the right is not.