Order of integration has never been my forte. I combines the two things I hate the most...shapes...and inequalities .

Ok so I don't understand what I am doing wrong here.

Change the order of integration

$\displaystyle \int_0^{\frac{\pi}{2}}\int_0^{\cos(x)}f(x,y)~dy~dx$

So

$\displaystyle 0\leq{x}\leq\frac{\pi}{2}$

and

$\displaystyle 0\leq{y}\leq\cos(x)$

$\displaystyle \Rightarrow\frac{\pi}{2}\leq\arccos(y)\leq{x}$

$\displaystyle \Rightarrow\arccos(y)\leq{x}\leq\frac{\pi}{2}$

and I can see that from the picture that the outer bound should be

$\displaystyle \int_0^{1}$

But could someone give me a quick run through on how to manipulate the inequalities?

I know the first part is wrong and the second right, but if you wouldn't mind doing both.