$$ \int \int_D (1-y) dxdy $$
$$ x^2+y^2 \leq 2y $$
$$ y \leq x^2 $$
$$ x \geq 0 $$
I have this exercise and I do not know how to solve it. I have to find the Domain for the x and the y coordinates. In the picture I can see a circle that is placed under a parabola and it is cutted in half vertically. Do you have to use polar coordinates? My region is the second picture.
$x^2 + y^2 = 2y$ can be written as $x^2 + (y-1)^2 = 1$ $\implies $ $y = 1 \pm \sqrt{1-x^2}$.
The parabola intersects the circle at $(0,0)$ and $( 1,1)$ in the right half plane.
so you can write \begin{align*} \int_D (1 - y) dy dx &= \int_{0}^{1} \int_{1 - \sqrt{1 - x^2}}^{x^2} (1 - y) dy \, dx\\ &= \int_0^1 \left(\frac{1}{2} x^2 - \frac{1}{2}x^4 \right) dx\\ &= \frac{1}{15} \end{align*}