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

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

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

$$-1le xle1 ,$$ and $$-sqrt{1-x^2}-e^xle ylesqrt{1-x^2}-e^x.$$