A curve is given by the function f(x)=6x−x2. Evaluate the area bounded by this curve and the x-axis, for x≥0!
Explanation
To calculate the area under the curve, first find the intersection points of the curve with the x-axis to determine the integration bounds.
Finding Intersection with the x-axis
The region is bounded by the curve f(x)=6x−x2 and the x-axis (y=0)
The intersection points of the curve with the x-axis are obtained when f(x)=0
6x−x2=0
x(6−x)=0
Therefore, x=0 or x=6
Determining Integration Bounds
Since the question asks for x≥0, the integration bounds are from x=0 to x=6
Calculating the Area
The area under the curve f(x) from x=0 to x=6 is calculated using a definite integral
Area=∫06(6x−x2)dx
Evaluating the Integral
∫(6x−x2)dx=3x2−31x3+C
Applying Integration Bounds
Area=[3x2−31x3]06
=[3(6)2−31(6)3]−[3(0)2−31(0)3]
=[3(36)−31(216)]−0
=[108−72]−0
=108−72
=36 square units
Therefore, the area bounded by the curve and the x-axis is 36 square units.