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 of the solid obtained by rotating the region bounded by the given curves about the x-axis, we can use the disk method. The disk method formula is V = π ∫ [R(x)]² dx from a to b, where R(x) is the radius of the disk at x.

1. Identify the curves: The region is bounded by y = x - 1, y = 0, and x = 8.

2. Identify the radius: The radius of the disk at x is the distance from the x-axis to the curve y = x - 1, which is simply the y-value of the curve. So, R(x) = x - 1.

3. Identify the limits of integration: The region starts at x = 1 (where y = x - 1 intersects y = 0) and ends at x = 8.

4. Plug into the formula and solve:

V = π ∫ from 1 to 8 (x - 1)² dx = π [ (1/3)x³ - x² + x ] from 1 to 8 = π [(1/3)(8)³ - (8)² + 8 - ((1/3)(1)³ - (1)² + 1)] = π [341/3 - 64 + 8 - (1/3 + 1 - 1)] = π [341/3 - 56 - 1/3] = π [285/3] = 95π cubic units.

So, the volume of the solid is 95π cubic units.