Based on the following given information answer the question
Consider the function \(f(x) = x|x|\), \(x \in \mathrm{R}\).
The area of the region bounded by \(f(x) = x|x|\), x-axis and from x = −1 to x = 1 is :
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(\displaystyle\frac{2}{3}\) sq. units
Here \(f(x) = x^{2}\) for \(x \ge 0\) (above the x-axis) and \(f(x) = -x^{2}\) for x < 0 (below the x-axis). An area below the x-axis is counted as positive, so the two parts are found separately.
From x = 0 to x = 1:
\( \displaystyle \int_{0}^{1} x^{2}\,dx = \frac{1}{3} \)
From x = −1 to x = 0:
\( \displaystyle \left|\int_{-1}^{0} -x^{2}\,dx\right| = \frac{1}{3} \)
Total area:
\( \displaystyle \frac{1}{3} + \frac{1}{3} = \frac{2}{3} \text{ sq. units} \)
Tip: \(\int_{-1}^{1} x|x|\,dx = 0\), because f is odd and the two parts cancel. That value, 0, is not the area; it is the trap in Option (2).
NCERT (2026-27 edition):
Application of Integrals
