Well, I think I'm switching study subjects after this, because I've about had my fill.
$\displaystyle 2\pi \int_{0}^{1} (1+x) \sqrt{1-x} dx$
I've tried about 10 different things, and can't seem to get anywhere.
Perhaps I'm doing it wrong:
Substitution definitions:
$\displaystyle u=1-x$
$\displaystyle \frac{du}{dx}=-1$
$\displaystyle -du=dx$
Origional equation:
$\displaystyle 2\pi \int_{0}^{1} (1+x) \sqrt{1-x} dx$
Substitute:
$\displaystyle 2\pi \int_{0}^{1} (1+x) \sqrt{u} (-1)du$
I still have 1+x leftover. But I am pretty tired now, I've been looking at these all day, I'm probably overlooking something obvious