Question

D is the planar region bounded by y = 0, y = x² and x = 1. Find the value of the integral below.​

Answer 1

$\displaystyle \iint_D x e^y \, dA = \int_0^1 \int_0^{x^2} x e^y \, dy \,dx \\\\ ~~~~~~~~ = \int_0^1 x (e^{x^2} - 1)\, dx \\\\ ~~~~~~~~ = \frac12 \int_0^1 e^u \, du - \int_0^1 x \,dx = \boxed{\frac{e - 2}2}$