How to calculate the double integral of $f(x,y)$ within the intersected area? $$f(x,y)=a_0+a_1y+a_2x+a_3xy$$ $a_0$, $a_1$, $a_2$, and $a_3$ are constants.
The area is the intersection of an ellipse and a circle.
Any help will be appreciated, I don't know how to do this.
Can I use $x=ra\cosθ$, $y=rb\sinθ$ to transform the ellipse and circle? If I can, what's the equation for the circle? What's the range for $r$ while calculating?
Let's say that the intersections are at $x=x_1$ and $x=x_2$. Also assume that the ellipse has equation $x^2/a^2 + y^2/b^2 = 1$ and the circle has equation $(x+c_x)^2 + (y+c_y)^2 = r^2$. The integral over the area then takes the form:
$$ \int_{x_1}^{x_2} dx \: \int_{-b \sqrt{1-x^2/a^2}}^{\sqrt{r^2 -(x+c_x)^2}-c_y} dy \: f(x,y) $$