How do I find the volume of a solid rotated around y = 3?

Question

I had a college textbook problem which states:

The region bounded by the curves y = x and y = x^2 is rotated about the line y = 3. Find the volume of the solid.

I know how to find the volume if it is not rotated by y = 3. But when it states rotated about the line y = 3. I have no idea how to do it.

Thanks for reading!

Answer 1

HINT

Very similar idea. Each vertical cross-section of the region in question, lying on some constant value $x$ has height of $h(x) = x - x^2$ and rotates in a circle of radius $3-x$. What is the circumference of the rotational circle? What is the surface area of the cylindrical slice generated at this $x$? Can you now compute the volume?

Answer 2

Does the above diagram help? See the shaded area that you are rotating around $y = 3$. Do you see that the radius you are interested in is between curves $y = x$ and $y = x^2$ and is function of $x$? So all you need to do is to find the limits of $r$ and set up the integral as $r \ dr \ dx \ d\theta$ and you know the limits of $x$.