Final answer:
To find the volume generated by rotating the region bounded by the curves y=11x-x² and y=28 about the x-axis using the method of cylindrical shells, follow these steps: find the limits of integration, determine the height of each cylindrical shell, find the radius of each cylindrical shell, and integrate the volume formula for cylindrical shells using the limits of integration.
Step-by-step explanation:
To find the volume generated by rotating the region bounded by the curves y=11x-x² and y=28 about the x-axis using the method of cylindrical shells, we can integrate the volume of concentric cylindrical shells.
- First, we need to find the limits of integration by setting the two equations equal to each other:
11x-x² = 28. Rearrange the equation to get a quadratic equation:
x²-11x+28 = 0. - Solve the quadratic equation to find the x-values at the points of intersection. The x-values will be the limits of integration.
- Next, we need to determine the height of each cylindrical shell. The height is the difference between the two equations, 28 - (11x-x²).
- The radius of each cylindrical shell is x.
- Finally, we integrate the volume formula for cylindrical shells:
V = ∫(2πx(28-(11x-x²)))dx. Integrate the formula using the limits of integration found in step 2 to calculate the volume.