Question

A region R in the first quadrant of the xy-plane is bounded by the curves y = x², y = 2 - x and the x-axis. Find the value of the following:​

Answer 1

The two curves meet in the first quadrant when

$x^2 = 2-x \implies x^2 + x - 2 = (x + 2) (x - 1) = 0 \implies x=1$

Then the integral in question is

$\displaystyle \iint_R x \, dA = \int_0^1 \int_{x^2}^{2-x} x \, dy \, dx \\\\ ~~~~~~~~ = \int_0^1 x (2-x-x^2) \, dx \\\\ ~~~~~~~~ = \int_0^1 (2x - x^2 - x^3) \, dx = 1 - \frac13 - \frac14 = \boxed{\frac5{12}}$