Questions on changing bounds of integration for double-integrals

Question

I'm having difficulty understanding how to properly change these bounds of integration. When I set up the bounds, which x or y bound is supposed to depend on the other?

$$\int_{0}^{\pi/2}\int_{0}^{\cos(y)}7\cos(y) dxdy$$

I came up with below, but I believe my method was flawed or I misunderstood which operations need to be swapped in order to properly set each minimum and maximum. I feel that I may be mixing up general trigonometric rules with setting up bounds.

$$\int_{0}^{\pi/2}\int_{\arccos(0)}^{\arccos(x)}7\cos(y) dydx$$

Answer 1

When integrating a function $f(x,y)$, $$\int_A^B\int_{C(y)}^{D(y)} f(x,y) dxdy$$ you can rewrite the integration region as $$ \{(x,y): A\le y \le B , C(y) \le x \le D(y) \}$$ For reasonable functions $C,D$, this is equivalent to $$ \{ (x,y) : \inf C \le x \le \sup D , D^{-1}(x)<y<C^{-1}(x)\}$$ which gets you $$\int_{\inf C}^{\sup D} \int_{D^{-1}(x)}^{C^{-1}(x)} f(x,y) dydx$$

Answer 2

$\int_{0}^{\cos(y)}7\cos(y) dx= 7 \cos^2(y)$,

hence

$ \int_{0}^{\pi/2}\int_{0}^{\cos(y)}7\cos(y) dxdy= 7 \int_{0}^{\pi/2} \cos^2(y) dy$.