Final answer:
To find the area under the curve y = -x² + 10x above the x-axis from x = 0 to x = 10, you solve the area by integrating the function. The end result is an area of approximately 166.67 square units.
Step-by-step explanation:
To find the area of the region bounded by the graph of the equation y = -x² + 10x and y = 0, you must first determine the points where the parabola intersects the x-axis. These points are found by setting the equation of the parabola equal to zero and solving for x:
−x² + 10x = 0
x(−x + 10) = 0
x = 0 or x = 10
These intersection points (0,0) and (10,0) define the limits of integration for calculating the area under the curve. Since the parabola opens downward, the area under the curve is above the x-axis and is therefore positive between these points.
To calculate the area, you integrate the function y = -x² + 10x with respect to x from 0 to 10:
∫ 0² (-x² + 10x) dx
Expanding the integrand gives:
∫ 0² (-x² + 10x) dx = ∫ 0² (−x³/3 + 5x²) | ⁰¹⁰
Computing the definite integral, we find out the area under the curve:
= [(−x³/3 + 5x²) | 10] − [(−x³/3 + 5x²) | 0] = (−1000/3 + 500) - (0) = 500 - 1000/3
= (1500− 1000)/3
= 500/3
The final result, which represents the area of the region bounded by the parabola and the x-axis, is approximately 166.67 square units.