Final answer:
To calculate the area between the curve f(x) = -6 + 5x - x^2 and the x-axis, we set up an integral with limits x1 and x2, which are the roots of the function. These points are found by solving the quadratic equation, and the integral calculates the net area between these points.
Step-by-step explanation:
To find the area between the graph of f(x) = -6 + 5x - x^2 and the x-axis, we need to setup the integral of f(x) over the interval where the function is above the axis or below it. First, we have to identify these intervals by finding the roots of the function where it crosses the x-axis. These roots can be found by setting the function equal to zero and solving for x:
0 = -6 + 5x - x^2
By solving the quadratic equation, we get the points of intersection, x1 and x2, which will be the limits of integration. The area A can be found using the integral:
A = ∫_x1^x2 (-6 + 5x - x^2) dx
This integral calculates the net area under the curve of f(x) between x1 and x2. If the resulting area is negative (when the graph is below the x-axis), we take the absolute value to get the actual area.