Polar Coordinates

Question

It's been ages since i did any coordinate conversions, and typically i have these two which i just can't manage to solve by myself.

1. I want to express the circle $x^{2}+y^{2}<4, x<0 $

2. The Area: $x-|y|\ge 0$

For the first problem i just thought it would be as easy as following:

$$0< r < 2$$

$$\frac {\pi}{2}<\theta<\frac{3\pi}{2}$$

For the second problem i don't even know how to begin...

Best Regards

Joe

Answer 1

For the second problem: $-\frac{\pi}{4} \leq \theta \leq \frac{\pi}{4}$ ($x \geq |y|$ hence $x$ is positive, $\cos\theta \geq |\sin\theta|$, from which the inequality for $\theta$ easily follows).

Answer 2

Maybe this will help you get started. Here's a picture of the region $x-|y|\geq0$:

$\hskip2in$

(generated here by Wolfram Alpha). From this picture, there's something you should be able to see immediately: will there be any dependence on $r$? In fact, can you work out what the region is from the picture?