Find the area enclosed by |x|+|y|=1?

Use |u| = u if u>= 0 and |u| = -u if u<0

So int the first quadrant |x| + |y| = 1 becomes x+y=1
and we now see this as a right angled issoceles triangle of sides 1, 1 and sqrt(2)
and angles pi/4, pi/4 and pi/2.

In the second quadrant |x|+|y|=1​ becomes -x + y =1 and this encloses a right angled issoceles triangle
​of sides 1, 1, and sqrt(2), with angles pi/4, pi/4 and pi/2

The same is true for the third and fourth quadrant.

So the region enclosed is a square of side sqrt(2).

So the area is (sqrt(2))^2 = 2

The diagonals of the square are of length 2, so the area is
.. (1/2)*2^2 = 2.

This is the interior of the square with vertices (0, ±1) and (±1, 0). This square has sides √2 and area 2

It's a square with sides equal to √2 so the area is (√2)² = 2