Integrals: Diagram-Based Questions (9)

Question

Find the area enclosed between the curves $y = x^2$ and $y = 2x - x^2$.

Solution — Step by Step

Why This Works

The area between two curves is the integral of (top minus bottom) over the interval where they enclose a region. The intersections give the limits, and a quick test point tells us which is which.

If the curves cross multiple times within the interval, we must split the integral at each crossing and possibly take absolute values — but here they only meet at the endpoints.

Alternative Method

Geometric symmetry: the curve $y = 2x - x^2 = 1 - (x-1)^2$ is a downward parabola with vertex at $(1, 1)$, and $y = x^2$ is an upward parabola through origin. The enclosed region is symmetric about $x = 1/2$ — we can compute the integral from $0$ to $1/2$ and double it for a quick check.

Common Mistake