Question

find the volumes of the solids generated by revolving the regions bounded by the graphs of the equations about the given lines. y = x2 y = 6x − x2 (a) x-axis

Answer 1

To find the volumes of the solids generated by revolving the regions bounded by the graphs of the equations about the x-axis, we can use the method of disk integration.

Step-by-step explanation:

To find the volumes of the solids generated by revolving the regions bounded by the graphs of the equations about the x-axis, we can use the method of disk integration. Here are the steps:

1. Find the points of intersection between the two curves y = x^2 and y = 6x - x^2 by setting them equal to each other and solving for x. The values of x obtained will be the bounds of integration.
2. For each value of x obtained, calculate the corresponding values of y for both curves.
3. For each value of x, calculate the radius of the disk by subtracting the y-coordinates of the two curves.
4. Using the formula for the volume of a disk, V = πr^2h, where r is the radius and h is the height (or thickness) of the disk, calculate the volume of each disk.
5. Finally, sum up the volumes of all the disks to obtain the total volume of the solid generated.