Let a > 0. If the area of the region bounded by the curve \(y = x^{2}\), the x-axis and the line x = a is equal to the area of the region bounded by the line y = x, the x-axis and the line x = a, then a is equal to
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\displaystyle\frac{3}{2}\)
Both curves lie above the x-axis for x > 0, so each area is the integral from 0 to a:
\( \displaystyle A_{1} = \int_{0}^{a}x^{2}\,dx = \frac{a^{3}}{3} \)
\( \displaystyle A_{2} = \int_{0}^{a}x\,dx = \frac{a^{2}}{2} \)
Setting the two areas equal:
\( \displaystyle \frac{a^{3}}{3} = \frac{a^{2}}{2} \)
\( \displaystyle a = \frac{3}{2} \) (dividing by \(a^{2}\), since a > 0)
Why the other options are incorrect
Option (1) 1 — Incorrect. This is where the curve and the line meet; there the areas are \(\displaystyle\frac{1}{3}\) and \(\displaystyle\frac{1}{2}\), not equal.
Option (3) \(\displaystyle\frac{2}{3}\) — Incorrect. The areas are \(\displaystyle\frac{8}{81}\) and \(\displaystyle\frac{2}{9}\), not equal.
Option (4) 3 — Incorrect. The areas are 9 and \(\displaystyle\frac{9}{2}\), not equal.