Mathematics

The key is recognizing that $y = |x|$ forms a V-shape while $y = x^2$ is a parabola, and they intersect at specific points where you can compare which function is above the other. By exploiting symmetry about the $y$-axis, you only need to integrate over half the interval, then double your result—this cuts the work in half and makes the absolute value disappear naturally.

The key is recognizing that $y = |x|$ forms a V-shape while $y = x^2$ is a parabola, and they intersect at specific points where you can compare which function is above the other. By exploiting symmetry about the $y$-axis, you only need to integrate over half the interval, then double your result—this cuts the work in half and makes the absolute value disappear naturally.