Area of $A:= \lbrace (x,y) \in \mathbb{R}^2 : x^2+(e^x +y)^2\leq1 \rbrace$

Question

I am having trouble calculating the area of $$A:= {\lbrace(x,y) \in \mathbb{R}^2 : x^2+(e^x +y)^2\leq1 \rbrace}.$$ I hope someone can help me.

I have tried using Fubini with the following boundaries for $x$ and $y$:

$$-1\le x\le1 ,$$ and $$-\sqrt{1-x^2}-e^x\le y\le\sqrt{1-x^2}-e^x.$$

Answer 1

Use Cavalieri's principle. The area is $\int_{-1}^12\sqrt{1-x^2}dx$, just like the circle we'd get without the $e^x$ term, i.e. $\pi$.