Evaluate ∫∫ e^(x²) dx dy

I = ∫₀¹ ∫_{3y}^{3} e^(x²) dx dy

e^(x²) has no elementary antiderivative, so we change the order and integrate with respect to y first.

Region

y runs from 0 to 1 and x from 3y to 3. The boundary lines are x = 3y (or y = x/3), x = 3 and y = 0. This is the triangle with vertices (0,0), (3,0), (3,1).

y
1 |       *(3,1)
  |     /|
  |   /  |   y = x/3
  | /____|__ x
 0        3

Taking vertical strips: x from 0 to 3 and y from 0 to x/3.

Reversed integral

I = ∫₀³ ∫₀^{x/3} e^(x²) dy dx
  = ∫₀³ e^(x²) (x/3) dx
  = (1/6) ∫₀³ 2x e^(x²) dx
  = (1/6) [ e^(x²) ]₀³
  = (e⁹ − 1)/6

Answer: I = (e⁹ − 1)/6 ≈ 1350.35.