The area (in sq. units) of the region bounded by the curve \(y=x^5\), the x-axis and the ordinates \(x = -1\) and \(x = 1\) is equal to
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\displaystyle\frac{1}{3}\)
The curve \(y = x^5\) lies below the x-axis for \(-1 \le x \lt 0\) and above it for \(0 \lt x \le 1\). Since \(x^5\) is an odd function, the two parts have equal areas:
\( \displaystyle \text{Area} = \int_{-1}^{1} |x^5|\,dx = 2\int_{0}^{1} x^5\,dx \)
\( \displaystyle = 2\left[\frac{x^6}{6}\right]_{0}^{1} = \frac{1}{3} \)
Why the other options are incorrect
Option (1) \(\displaystyle\frac{1}{6}\) — Incorrect. This is the area on one side of the y-axis only; the part between x = −1 and x = 0 adds another \(\frac{1}{6}\).
Option (3) \(\displaystyle\frac{1}{2}\) — Incorrect. The area is \(\frac{1}{3}\), not \(\frac{1}{2}\).
Option (4) \(\displaystyle\frac{2}{3}\) — Incorrect. The area is \(\frac{1}{3}\), not \(\frac{2}{3}\).