Let $f(x)$ be continuous on $[0,1]$ with $\int_0^1 f(x)dx=\alpha$. Find $\int_0^1 \int_x^1 f(x)f(y)dydx$.
I don't really know what I'm doing at all with this one. I started off by letting $F(x)$ be an antiderivative of $f$, and working out the inner integral as $\int_x^1 f(x)f(y)dy=f(x)(F(1)-F(x))$. But that's the only thing I can come up with, and I think it doesn't really get anywhere, because I then have to integrate that expression with respect to x, which doesn't seem possible.