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

Question

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 $$-sqrt{1-x^2}-e^xle ylesqrt{1-x^2}-e^x.$$

J.G. · Accepted Answer

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

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

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