The area bounded by the curve $y = x|x|$, x-axis and the ordinates $x = -1 \text{ and } x = 1$ is given by

Concept: Area (not the signed integral) under a piecewise curve.

For $x \ge 0, y = x|x| = x^2$. For $x < 0, y = x|x| = -x^2$, which lies below the x-axis. Area, being always non-negative, requires taking the absolute value of the curve on each sub-interval.

• $\text{Area from } x=-1 \text{ to } 0 = \int |-x^2|\,dx = \int x^2\,dx = \frac{1}{3}$
• $\text{Area from } x=0 \text{ to } 1 = \int x^2\,dx = \frac{1}{3}$
• $\text{Total area} = \frac{1}{3}+\frac{1}{3} = \frac{2}{3}$

Why the distractors fail:

• (A) 0 results from wrongly computing the signed definite integral of x|x| directly (which is 0 because x|x| is odd) and mistaking that for the geometric area — a very common conceptual trap in 'area bounded by curve' questions.
• (B) 1/3 accounts for only one of the two symmetric pieces (e.g. only x in [0,1]), forgetting the mirror region on the negative side.
• (D) 4/3 typically comes from doubling incorrectly (e.g. using $4 \times \frac{1}{3}$ by miscounting sub-regions).

Exam tip: Whenever a question says 'area bounded by curve and x-axis,' always split at the x-axis crossings and take the absolute value of the integral on each piece — never integrate straight through when the curve changes sign.