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- #1

\pi \int_{0}^{\pi/2}\cos ^2xdx

i thought this meant that the area function is

pi \cos ^2xdx and since \cos ^2xdx = 1^2 - \sinx ^2xdx

it's the solid obtained by rotating region between y = sinx and y = 0 for 0 <= x <= pi / 2.

but this was wrong.

how?

#2 compute the volume of a torus which has inner radius R and outer radius R + r.

no idea.