Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line.y = x − 1,  y = 0,  x = 8;    about the x-axisV = 95π

To find the volume of the solid obtained by rotating the region bounded by the curves y = x - 1, y = 0, and x = 8 about the x-axis, we can use the method of cylindrical shells.

Step 1: Identify the region bounded by the curves. The region is a right triangle with vertices at (1,0), (8,0), and (8,7).

Step 2: Set up the integral for the volume of the solid. The volume V of the solid is given by the integral from a to b of 2π times the radius times the height of the cylindrical shell. The radius is y and the height is x - 1. So, we have:

V = ∫ from 0 to 7 of 2πy(x - 1) dy

Step 3: Evaluate the integral. First, distribute the y inside the parentheses:

V = ∫ from 0 to 7 of 2πyx - 2πy^2 dy

Then, split the integral into two parts:

V = 2π ∫ from 0 to 7 of yx dy - 2π ∫ from 0 to 7 of y^2 dy

Since x is a constant with respect to y, the first integral simplifies to:

2πx ∫ from 0 to 7 of y dy = 2πx * [1/2 y^2] from 0 to 7 = 2πx * 49/2 = 49πx

The second integral is a standard power rule integral:

2π * [1/3 y^3] from 0 to 7 = 2π * 343/3 = 686π/3

So, the total volume is:

V = 49πx - 686π/3

Step 4: Substitute x = 8 into the volume equation. V = 49π*8 - 686π/3 = 392π - 686π/3 = 95π

So, the volume of the solid obtained by rotating the region bounded by the curves about the x-axis is 95π cubic units.