change the order of integration for the following integral from dydx to dxdy, and from dydx to polar coordinates. $$ \int \int f(x,y) dydx$$
where $$ 0≤y≤(-x^2)+2 $$ $$ 0≤x≤1$$
From dydx to dxdy $$ \int \int f(x,y) dxdy + \int \int f(x,y) dxdy$$
$$ 0≤x≤1 $$ $$ 0≤y≤1 $$
Second integral $$ 0≤x≤\sqrt(2-y) $$ $$ 0≤y≤1 $$
I'm not sure about the $\sqrt(2-y)$ for the bounds of x for the second integral.
I'm having more trouble converting this into polar coordinates though. I think I can leave the first integral as it is in terms of dxdy, because the region is a rectangle. Is there any way to switch this rectangular region into polar coordinates?
For the second integral
$$ 0≤r≤1 $$ $$ 0≤\theta≤\pi/2 $$
$$ \int \int f(x,y) dxdy + \int \int f(r,\theta) rdrd\theta$$