The area (in sq. units) of the region bounded by the curve \(y = 6x-x^{2}\) and the x-axis is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 36
The curve meets the x-axis where
\(6x - x^{2} = 0\)
\(x(6-x) = 0\), so x = 0 and x = 6
Between these points y > 0 (at x = 3, y = 9), so the curve is above the x-axis and the area is
\( \displaystyle A = \int_{0}^{6}\left(6x-x^{2}\right)dx \)
\( \displaystyle = \left[3x^{2}-\frac{x^{3}}{3}\right]_{0}^{6} \)
\(= 108 - 72 = 36\)