Hi,

I have the following prblem:

Let $\displaystyle m_2$ be the Lebesgue measure on $\displaystyle \mathbb{R} ^2$.

(i) Given $\displaystyle c\in\mathbb{R}$ find the measure of the set $\displaystyle \{(x,y)\in\mathbb{R}^2; x+y=c\}$.

My solution: I simply drew the set and jcomputed the length using the Pythagorean Thm, so $\displaystyle m_2(\{(x,y)\in\mathbb{R}^2; x+y=c\})=\sqrt 2 |c|$.

(ii) Find $\displaystyle m_2(A)$, where $\displaystyle A$ consists of all pairs $\displaystyle (x,y)$ in $\displaystyle [0,\pi/2]\times[0,\pi/2]$ such that $\displaystyle \cos (x)\ge 1/2$ and $\displaystyle \sin(y)$ is irrational.

My attempt: I tried to do the same thing here, but as you can see the set is a bit more complicated. This is what I've got so far: since $\displaystyle \cos (x)\ge 1/2$ iff. $\displaystyle x\le\pi/3$ we have $\displaystyle 0\le x\le \pi/3$.

What next?