Use plane polar co-ordinates or otherwise to evaluate the integral
$$\int\int_D^\ \frac{x^2-y^2}{x^2+y^2} dA$$
where D is the part of the x,y plane bounded by the parabola $y^2=4(1-x)$ and the coordinate axes in the positive quadrant. In plane co-ordinates the parabola has the equation $r= \frac{2}{1+cos(\theta)}$
I'm not sure how to find the limits for this problem. Could someone point me in the right direction?
$\int\limits_{x=0}^1 \int\limits_{y=0}^{2 \sqrt{1-x}} {x^2 - y^2 \over x^2 + y^2} dx\ dy = 2 \pi -\frac{20}{3}$
Clearly, the variable $x$ has the range $0 \le x \le 1$. For any value of $x$, the curve shows that $0 \le y \le \sqrt{4(1-x)} = 2 \sqrt{1-x}$.