alculate the volume of the solid of revolution that is formed by revolving the curvey = 2 + sin(x) over the interval [0, 2π] around the x-axis.

To calculate the volume of the solid of revolution, we can use the disk method. The formula for the disk method 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 radius function r(x). In this case, the curve y = 2 + sin(x) is being revolved around the x-axis, so the radius at any point x is just the y-value at that point, which is 2 + sin(x).

2. Plug r(x) into the formula and set up the integral. We get V = π ∫ [(2 + sin(x))]² dx from 0 to 2π.

3. Simplify the integrand. (2 + sin(x))² = 4 + 4sin(x) + sin²(x).

4. Now we have V = π ∫ (4 + 4sin(x) + sin²(x)) dx from 0 to 2π.

5. To integrate sin²(x), we can use the power-reduction identity, which says that sin²(x) = 1/2 - 1/2cos(2x). So the integral becomes V = π ∫ (4 + 4sin(x) + 1/2 - 1/2cos(2x)) dx from 0 to 2π.

6. Now we can integrate term by term. The integral of 4 is 4x, the integral of 4sin(x) is -4cos(x), the integral of 1/2 is 1/2x, and the integral of -1/2cos(2x) is -1/4sin(2x).

7. Evaluate each of these from 0 to 2π. For 4x, we get 42π - 40 = 8π. For -4cos(x), we get -4cos(2π) - (-4cos(0)) = 0. For 1/2x, we get 1/22π - 1/20 = π. For -1/4sin(2x), we get -1/4sin(4π) - (-1/4sin(0)) = 0.

8. Add these up to get the total volume. V = π * (8π + 0 + π + 0) = 9π² cubic units.