Originally Posted by

**angel.white** 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:

$\displaystyle V = \pi \int_0^1((2x-x^2)^2 -(x^3)^2)~dx + \pi\int_{-2}^0((x^3)^2 -(2x-x^2)^2) ~dx$

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)