TITLE: Area under $ |x| + |y| < k $
QUESTION [5 upvotes]: Consider the region $C$ of all points in the $x,y$ plane that satisfy $ |x| + |y| < k $. Now I wish the find out the area of $C$ using an integral. It's easy all points in the interior of a square of side $k\sqrt{2}$ centered at the origin and tilted by $\pi/4$ satisfy the equation. The image below illustrates the region for $k=6$

So I guess that the area of $C$ is $2k^2$. I wasn't able to find this by using integral so I'd like an approach by using integrals. Any help would be appreciated. Thanks.

REPLY [2 votes]: You could move to a new coordinate system $(s,t)$ with $s=x+y$, $t=x-y$.
Then $dA=\begin{vmatrix}1&1\\1&-1\end{vmatrix}ds\,dt=2\,ds\,dt$.
In that coordinate system, $$A=\int_{-k/2}^{k/2}\int_{-k/2}^{k/2}kdA=\int_{-k/2}^{k/2}\int_{-k/2}^{k/2}2\,ds\,dt=2k^2$$