Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line.y = x,   y = 0,   x = 2,   x = 6;  about x = 1

To find the volume of the solid obtained by rotating the region bounded by the curves y = x, y = 0, x = 2, and x = 6 about the line x = 1, we can use the method of cylindrical shells.

Step 1: Identify the region The region is a rectangle with vertices at (2,2), (2,0), (6,6), and (6,0).

Step 2: Set up the integral The formula for the volume of a solid of revolution using cylindrical shells is V = 2π ∫ [a,b] (x - h)f(x) dx, where h is the axis of rotation. Here, a = 2, b = 6, h = 1, and f(x) = x.

Step 3: Evaluate the integral V = 2π ∫ [2,6] (x - 1)x dx = 2π [ (1/2)x^2 - x | from 2 to 6] = 2π [(1/2)(6)^2 - 6 - (1/2)(2)^2 + 2] = 2π [18 - 6 - 2 + 2] = 2π [12] = 24π cubic units

So, the volume of the solid obtained by rotating the region bounded by the curves about the line x = 1 is 24π cubic units.

To find the volume of the solid obtained by rotating the region bounded by the curves y = x, y = 0, x = 2, and x = 6 about the line x = 1, we can use the method of cylindrical shells.

The formula for the volume of a solid of revolution using cylindrical shells is V = ∫[a,b] 2πrh dx, where r is the radius of the shell and h is the height of the shell.

Step 1: Identify the radius and height of the shells. The radius of the shell is the distance from the shell to the line of rotation, which is |x - 1|. The height of the shell is the value of the function at x, which is y = x.

Step 2: Set up the integral. The limits of integration are the x-values where the region is bounded, which are x = 2 and x = 6. So the integral is V = ∫[2,6] 2π|x - 1|x dx.

Step 3: Evaluate the integral. This is a simple integral to evaluate. First, simplify the integrand to 2π(x - 1)x. Then, distribute the x to get 2π(x^2 - x). Now, integrate term by term to get V = 2π[(1/3)x^3 - (1/2)x^2] evaluated from 2 to 6.

Step 4: Substitute the limits of integration. V = 2π[(1/3)(6)^3 - (1/2)(6)^2] - 2π[(1/3)(2)^3 - (1/2)(2)^2] = 2π[72 - 18] - 2π[8/3 - 2] = 2π[54 - 2/3] = 2π[161/3] = 322π/3 cubic units.

So, the volume of the solid obtained by rotating the region bounded by the curves about the line x = 1 is 322π/3 cubic units.