Surely, you were given some instructions in your textbook or class materials. Give it a go.
[math]\int_{0}^{2}\sqrt{What?}\;dy[/math]
Tell us that and we can talk about evaluating the integral expression.
Have you considered transforming the problem statement to y = f(x), first? (Rather than x = g(y))
Now you have to integrate [1 +(e-y)2]1/2dyinside the radical I have 1+(-e^(-y))^2