The area (in sq. units) of the region bounded by the curve y = x|x|, the x-axis and the lines x = −1 and x = 2 is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 3
Opening the modulus:
\(x \ge 0\): \(y = x^{2}\) (above the x-axis)
\(x \lt 0\): \(y = -x^{2}\) (below the x-axis)
The part below the x-axis is counted with a positive sign, so the area is found in two parts:
\( \displaystyle \int_{-1}^{0}(-x^{2})\,dx = -\frac{1}{3} \), so this part has area \(\displaystyle\frac{1}{3}\)
\( \displaystyle \int_{0}^{2}x^{2}\,dx = \frac{8}{3} \)
Adding the two parts:
\( \displaystyle A = \frac{1}{3}+\frac{8}{3} = 3 \)
Why the other options are incorrect
Option (2) \(\displaystyle\frac{7}{3}\) — Incorrect. This is the single integral from −1 to 2, \(\displaystyle -\frac{1}{3}+\frac{8}{3}\); the part below the axis is subtracted instead of added.
Option (3) \(\displaystyle\frac{8}{3}\) — Incorrect. This is only the part from x = 0 to x = 2.
Option (4) \(\displaystyle\frac{1}{3}\) — Incorrect. This is only the part from x = −1 to x = 0.