Define a region to be the points such that

Sketch this region, and compute the volume of the solid of revolution under the following revolutions:

- Revolution about the -axis.
- Revolution about the -axis.
- Revolution about the vertical line through the point .
- Revolution about the horizontal line through the point .

First, the sketch of the region is:

- We compute the volume from revolving about the -axis as follows,
- First, we find an equation for in terms of .
Then, we compute the volume as follows from revolving about the -axis as follows,

- We compute the volume as follows from revolving about the vertical line as follows,
- We compute the volume as follows from revolving about the horizontal line as follows,

In part (d), shouldn’t one compute the integral of (1-1/4x²)² instead of (1/4x²-1)²? I know it leads to the same result, but it makes more sense to me, once one shifts the axis forward. Also, is it okay to rotate the figure around an axis that intercept it without subtracting anything? I mean, the solid will intercept itself when one rotate the figure, so one would be integrating some part of the figure twice.

I am not really clear on this question’s last two parts. Can you please help me. I really need this

Thank you

got it dude. !!

How would you attempt this: find the centroids of the solids formed by rotating completely about the x-axis the plane regions defined by y > x(x-4), y<0?