TITLE: Solid of revolution answer verification
QUESTION [0 upvotes]: The question is:
Does the integral $ \int _0^2 \pi(2^2-(2-y)^2)dy$ represent the volume of hemisphere or a cone. Give the radius of the hemisphere or radius and height of the cone.
The solution given to me:
This integral represents the volume of a cone with radius 2 and height 2. However, I don’t see how they got this answer.
My solution:
This integral represents $2/3$ of the volume of a right cylinder with radius 2 and height 2. I even confirmed this by evaluating the given integral as is and it equals $2/3$ the volume of a right cylinder with radius 2 and height 2. So it’s a right cylinder with a cone removed from it.
Any help with this would be very much appreciated!

REPLY [1 votes]: HINT:

The full cylinder has diameter 4 and height 2;
Between $ x_1 =2,  x_2=2-y $ the volume is obtained by rotating about $y$- axis. It is the outer red shaded volume got after removing cone from cylinder. (Has double the volume of cone).
$$V= \int(\pi x_1^2-\pi x_2^2) dy$$
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