Area of $\{(x,y)^T\in\mathbb{R}^2\mid x^2+y^2<r^2 \wedge -|x|<y<|x|\}$ What is the area of $\{(x,y)^T\in\mathbb{R}^2\mid x^2+y^2&lt;r^2 \wedge -|x|&lt;y&lt;|x|\}$?
I have that $x\in[0,r)$ but what about $y$. There I have two restrictions.
First of all x is not in [0,r] but in [-r,r].You will have to see x in 4 cases.
i)$x\in [-r,-r/\sqrt 2]$ then $y^2&lt;r^2-x^2$
ii)$x\in [-r/\sqrt 2),0]$ then x&lt;y&lt;-x
iii)$x\in [0,r/\sqrt 2),]$ then -x&lt;y&lt;x
iv)$x\in [r/\sqrt 2),r]$ then $y^2&lt;r^2-x^2$
First of all x is not in [0,r] but in [-r,r].You will have to see x in 4 cases.
i)$x\in [-r,-r/\sqrt 2]$ then $y^2&lt;r^2-x^2$
ii)$x\in [-r/\sqrt 2),0]$ then x&lt;y&lt;-x
iii)$x\in [0,r/\sqrt 2),]$ then -x&lt;y&lt;x
iv)$x\in [r/\sqrt 2),r]$ then $y^2&lt;r^2-x^2$