Final answer:
The volume of the solid formed by rotating the curve y = -x² + 13x - 40 around the x-axis is found by solving the integral π∫ ((-x² + 13x - 40)²) dx over the x-values where the curve intersects the x-axis.
Step-by-step explanation:
To find the volume of the solid of revolution formed by rotating the region bounded by the curve y = -x² + 13x - 40 and the x-axis about the x-axis, we can use the method of discs/rings. This is a calculus problem which involves setting up an integral.
First, we need to find the points of intersection between the curve and the x-axis to determine the limits of integration. Setting y to zero gives -x² + 13x - 40 = 0. Solving this quadratic equation yields the limits of integration as the roots of the equation.
Once the limits are found, we can use the disc method. The volume V is given by the integral from a to b of π multiplied by the square of the function defining the radius, which in this case is -x² + 13x - 40. The integral will be of the form V = π ∫ab ((-x² + 13x - 40)²) dx.
After setting up the integral, we evaluate it over the interval from the lower to the upper limit which gives the volume of the solid.