The area (in sq. units) of the region bounded by the curve \(y = (x-2)^{2}\), the x-axis and the y-axis is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(\displaystyle\frac{8}{3}\)
The curve is a parabola with vertex (2, 0), touching the x-axis at x = 2, and it meets the y-axis at (0, 4). So the region lies above the x-axis from x = 0 to x = 2:
\( \displaystyle A = \int_{0}^{2}(x-2)^{2}\,dx \)
\( \displaystyle = \left[\frac{(x-2)^{3}}{3}\right]_{0}^{2} \)
\( \displaystyle = 0-\left(-\frac{8}{3}\right) = \frac{8}{3} \)