The area of the region bounded by the curve \(x=y^2\) and the line x = 4 is equal to:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\displaystyle\frac{32}{3}\) square units
The parabola \(x = y^2\) opens to the right and is symmetric about the x-axis; it meets the line x = 4 at (4, 2) and (4, −2).
The part above the x-axis is \(y = \sqrt{x}\), from x = 0 to x = 4. The required area is twice this part:
\( \displaystyle \text{Area} = 2\int_{0}^{4}\sqrt{x}\,dx \)
\( \displaystyle = 2\left[\frac{2}{3}x^{3/2}\right]_{0}^{4} \)
\( \displaystyle = 2 \times \frac{2}{3} \times 8 \)
\( \displaystyle = \frac{32}{3} \text{ square units} \)