The area (in sq. units) of the region bounded by the curve x = y^2 ,…

The correct answer is Option (1) → \(\displaystyle\frac{37}{3}\)

The region lies between the y-axis (x = 0) and the curve \(x = y^{2}\), from y = 3 to y = 4. So the area is found by integrating x with respect to y:

    \( \displaystyle A = \int_{3}^{4} y^{2}\,dy \)

    \( \displaystyle = \left[\frac{y^{3}}{3}\right]_{3}^{4} \)

    \( \displaystyle = \frac{64-27}{3} = \frac{37}{3} \)

Why the other options are incorrect

• Option (2) \(\displaystyle\frac{47}{3}\) — Incorrect. The integral is \(\frac{64-27}{3} = \frac{37}{3}\), not \(\frac{47}{3}\).

• Option (3) 7 — Incorrect. This is \(4^{2} - 3^{2}\), the change in \(y^{2}\), not its integral.

• Option (4) \(\displaystyle\frac{91}{3}\) — Incorrect. This adds the two values, \(\frac{64+27}{3}\), instead of subtracting the lower limit value.