Question

6 find the area of the region bounded by the parabolas $y = 2x - x^2$ and $y = x^2$.

Explanation:

Step1: Find Intersection Points

Set \(2x - x^2 = x^2\). Simplify: \(2x - 2x^2 = 0 \Rightarrow 2x(1 - x)=0\). Solutions: \(x = 0\) or \(x = 1\).

Step2: Set Up Integral

The upper function is \(y = 2x - x^2\), lower is \(y = x^2\). Area \(A=\int_{0}^{1}[(2x - x^2)-x^2]dx=\int_{0}^{1}(2x - 2x^2)dx\).

Step3: Integrate

Antiderivative: \(x^2-\frac{2}{3}x^3\). Evaluate from 0 to 1: \((1 - \frac{2}{3})-(0 - 0)=\frac{1}{3}\).

Answer:

\(\frac{1}{3}\)