I need some help with this double integral?

Question

I need to find .... the double integral of 2y*e^(x^2)dydx
It's in the first quadrant and bounded by x=y^2 x=2 y=0 Thank you in advance

Answer 1

Substitute y=Sqrt[x] for x = y^2. Then your integral is of the form

double integral of 2y*Exp[x^2] dy dx. with the first integral from 0 to sqrt(x), and the second from 0 to 2.

The answer is (e^4 -1) / 2.

Steve

Answer 2

You have
2 √x
∫ .. ∫2y*e^(x^2)dydx =
0 0

2
∫ x*e^(x^2)dx =
0

let u = x^2, du = 2xdx

4
∫ (1/2)e^(u)du =
0

(1/2)(e^4 - e^0) =
(1/2)(e^4 - 1) = 26.799 ≈ 26.8