Question

The region bounded by the curves y = -x² + 11x - 30 and y = 0 is rotated about the x-axis. Find the volume V of the resulting solid by any method.

Answer 1

The volume of the solid formed by rotating the region bounded by the curve y = -x² + 11x - 30 and y = 0 about the x-axis can be found by using the disk method, which involves setting y = 0 to find the points of intersection, determining the limits of integration, and integrating the square of the function multiplied by pi across those limits.

Step-by-step explanation:

To find the volume V of the solid formed by rotating the region bounded by the curve y = -x² + 11x - 30 and the line y = 0 about the x-axis, we can use the method of cylindrical shells or the disk method. Since the region is between the curve and the x-axis, the disk method is more suitable here.

First, we find the points of intersection for the curve and the x-axis by setting y = 0 and solving the quadratic equation:

• -x² + 11x - 30 = 0

This will give us the limits of integration for the volume calculation. Next, we use the formula for the volume of a solid of revolution (disk method), which is

V = π ∫ (radius of disk)² dx

In this case, the radius of the disk is simply the value of the function y = -x² + 11x - 30, and x varies between the points of intersection found earlier. Integrating this expression from the lower to the upper limit will give us the volume of the solid