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 =

To find the volume V 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 disks or washers.

The formula for the volume of a solid of revolution using the disk method is V = ∫[a,b] π[f(x)]² dx, where [a,b] is the interval over which we're rotating, f(x) is the function we're rotating, and dx indicates that we're rotating with respect to x.

Here, our function f(x) is y = x - 1, and we're rotating over the interval [1,8] (since y = 0 when x = 1 and we're given x = 8 as the other bound).

So, we have:

V = ∫[1,8] π[(x - 1)]² dx = π ∫[1,8] (x² - 2x + 1) dx = π [x³/3 - x² + x] from 1 to 8 = π [(8³/3 - 8² + 8) - (1³/3 - 1² + 1)] = π [(512/3 - 64 + 8) - (1/3 - 1 + 1)] = π [(512/3 - 56) - (1/3)] = π [(512 - 168)/3 - 1/3] = π [(344/3) - 1/3] = π [343/3] = 343π/3 = 114.33π cubic units.

So, the volume V of the solid obtained by rotating the region bounded by the curves about the x-axis is 114.33π cubic units.