Area included in the curves

Question

What is the area included in the curves

√x + √|y| = 1 and |x| + |y| = 1.

I know the area the total area of the |x| + |y| = 1 is 2 units square but i cannot determine the area of the given first curve .. Please help.

Answer 1

Hint: Since $$x\geq 0$$ we get $$\sqrt{x}+\sqrt{|y|}=1$$ and $$x+|y|=1$$ and you can compute $y$

Answer 2

The branch of $\sqrt{|x|}+\sqrt{|y|}=1$ in the first quadrant is $$ y=(1-\sqrt{x})^2=1-2\sqrt{x}+x $$ The area under it is $$ \int_0^1(1-2\sqrt{x}+x)\,dx $$ The area under the line $x+y=1$ (from $x=0$ to $x=1$) is…

Subtract and multiply by $4$.