Question

Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the curves y = 5, x = 0, y = 5, y = 7 about the x = 5.

Answer 1

The volume of the solid is found using cylindrical shells by integrating the formula 2πrh, where 'r' is the distance from the rotation axis (x=5) to the bounded region and 'h' is the difference in y-values (y=7 minus y=5).

Step-by-step explanation:

The question involves the use of cylindrical shell method to determine the volume of a solid of revolution. To solve the problem, we need to visualize the region bounded by y=5, y=7, and x=0 being revolved around the line x=5. Using the formula for the volume of cylindrical shells (V = 2πrh), we arrive at the integral setup:

V = ∫₀⁷ 2π (5 - x) (7 - 5) dx

Here, (5 - x) acts as the radius (r) and (7 - 5) as the height (h). We can integrate from x=0 to x=7 since these are the bounds for y=5 and y=7.

After performing this integration, we would get the volume of the solid.