Let R be the region in the first quadrant bounded by the curves y = x^3 and y = 2x-x^2. Calculate the following quantities.
(a) The area of R
(I did this already and got 37/12 units squared)
(b) The volume obtained by rotating R about the x-axis
(Having trouble with this one. I feel like I know what I'm doing, but can't seem to get a satisfactory answer. I split it into two integrals:
But the second integral keeps returning negative volume, and I don't understand why. I am pretty sure I put the correct function on top, and that I took it over the correct interval. :/ I could just switch the sign (and hope that I did everything else correctly, it's an even problem so don't know the answer) but I wouldn't have any idea why I was doing that other than the method I thought should work didn't.
(c) The volume obtained by rotating R about the y-axis
(haven't even attempted this yet, can't get past b)