Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the line x = 6.

y = 6 − x
y = 0
y = 5
x = 0

You can use shells:

The region has to be broken up into the constant-height region where 0<=x<=1 and the slanting region where 1<=x<=6:

V = 2pi*Int(r*5 dx)[0,1] + 2pi*Int(rh dx)[1,6]
where r = 6-x
h = 6-x
= 138 2/3 pi

Or, using discs,

V = pi*Int(6^2 - y^2) dy[0,5]
= 138 2/3 pi

To find the volume of the solid generated by revolving the region bounded by the graphs of these equations about the line x = 6, we can use the method of cylindrical shells.

The region bounded by these equations forms a triangular region in the first quadrant of the xy-plane, with vertices at (0,0), (5,0), and (0,5). This region is the base of the solid.

To find the height of each cylindrical shell, we need to find the difference between the y-coordinates of the upper and lower curves at each x-value along the base.

From the equations, we can see that the upper curve is given by y = 6 - x, and the lower curve is y = 0.

When we revolve the region about the line x = 6, the x-coordinate of each cylindrical shell will be the distance from the line x = 6 to the corresponding x-value on the base.

Let's break down the steps to find the volume:

1. Determine the limits of integration: We need to find the x-values where the curves intersect. From the equations, we can set y = 6 - x and y = 0 equal to each other:
6 - x = 0
x = 6
So, the limits of integration for x are 0 to 6.

2. Find the height of each cylindrical shell: The height of each shell is the difference between the upper and lower curves at each x-value. In this case, it is (6 - x) - 0 = 6 - x.

3. Find the radius of each cylindrical shell: The radius of each shell is the distance from the line x = 6 to the corresponding x-value on the base. This distance is 6 - x.

4. Set up the integral: The volume of each cylindrical shell can be calculated by multiplying its height, radius, and the thickness of the shell (dx). So, the integral that represents the volume is:
V = ∫[0 to 6] (2π(6 - x)(6 - x) dx)

5. Evaluate the integral: Calculate the integral to find the volume of the solid.
V = ∫[0 to 6] (2π(6 - x)^2 dx)

After evaluating the integral, you can find the volume of the solid generated by revolving the given region about the line x = 6.