Step 1: Understanding the Concept:
The function $y = x|x|$ is defined as $x^2$ for $x \geq 0$ and $-x^2$ for $x<0$. Area is calculated as $\int |y| dx$ because area cannot be negative. Step 2: Formula Derivation:
Area = $\int_{-1}^{0} |-x^2| dx + \int_{0}^{1} |x^2| dx$
Area = $\int_{-1}^{0} x^2 dx + \int_{0}^{1} x^2 dx$ Step 3: Explanation:
$$\text{Area} = \left[ \frac{x^3}{3} \right]_{-1}^{0} + \left[ \frac{x^3}{3} \right]_{0}^{1}$$
$$\text{Area} = (0 - (-1/3)) + (1/3 - 0) = 1/3 + 1/3 = 2/3$$ Step 4: Final Answer:
The total area is $2/3$.