Find the volume of the solid formed by revolving the region bounded by y=1-x, y=x-4, x=0 about x=7

To find the volume of the solid formed by revolving the given region about the line x=7, we can use the method of cylindrical shells.

First, let's consider the region bounded by the curves y=1-x, y=x-4, and x=0. Sketching this region will visualise its shape.

To find the limits of integration, we need to determine the x-values at which the curves intersect.

Setting 1-x = x-4, we can solve for x:

1 - x = x - 4
2x = 5
x = 5/2

So, the curves y=1-x and y=x-4 intersect at x = 5/2.

Next, we need to determine the height of each cylindrical shell. This can be determined by subtracting the values of the curves at a specific x-value from the axis of rotation x=7.

Let's take an arbitrary value of x and calculate the height of the corresponding shell.

Let x be any value between 0 and 5/2.

At x = t, the height of the shell is given by:
h(t) = 7 - (1-t) = 6 + t

To find the volume of the solid formed by revolving the region, we integrate the product of the height of the cylindrical shell and the circumference of the shell over the range of x from 0 to 5/2:

V = ∫[0, 5/2] of 2πr*h(t) dt
= ∫[0, 5/2] of 2π * (7-t) * (6+t) dt

Evaluating this integral will give you the volume of the solid.