Question

The area bounded by the curve \(y = x|x|\), X-axis and the ordinates \(x = -1\) and \(x = 1\) is _____

• A. \( 0 \)
• B. \( \frac{2}{3} \)
• C. \( \frac{1}{3} \)
• D. \( \frac{4}{3} \)

Hint:

For area problems, always consider modulus or sign changes and split intervals accordingly.

Answer 1

The Correct Option is B

Concept: For modulus functions: \[ x|x| = \begin{cases} x^2, & x \ge 0 \\ - x^2, & x<0 \end{cases} \] Area is always taken positive.
Step 1: Split the interval. \[ \text{Area} = \int_{-1}^{0} -x^2 dx + \int_{0}^{1} x^2 dx \]
Step 2: Evaluate integrals. \[ \int_{-1}^{0} -x^2 dx = \frac{1}{3}, \quad \int_{0}^{1} x^2 dx = \frac{1}{3} \]
Step 3: \[ \text{Total Area} = \frac{1}{3} + \frac{1}{3} = \frac{2}{3} \]