I have to calculate the double integral in a 'D' domain: $$ D= \{x^2 + y^2 \le 1; x^2 +(y-1)^2 \ge 1 ; x^2 + (y+1)^2 \ge 1 \}$$
I want to write this domain in polar coordinates: I found myself this. $$ 0 \le r \le 1$$ $$ r\ge 2sin(theta) $$ $$ r \ge -2sin(theta) $$
the integral is : $$ \iint (1-x) dxdy $$ over the domain D; my question is: how could I write a form like $$ something \le theta \le something $$ PS: I already have r, if it's correct.
The integral is symmetric respect to $x$ axis, then \begin{align} \iint (1-x) dxdy &= 2\int_{y\geq0}\int (1-x) dxdy \\ &= 2\int_0^{\pi/6}\int_{2\sin\theta}^1(1-r\cos\theta)r\ dr\ d\theta+2\int_{5\pi/6}^\pi\int_{-2\sin\theta}^1(1-r\cos\theta)r\ dr\ d\theta \\ &= \color{blue}{\dfrac16+\sqrt{3}-\dfrac{\pi}{3}} \end{align}