Question(1 point) Find the volume of the solid that results when the region bounded by and is revolved about the line . Volume
Studdy Solution
STEP 1
1. The region is bounded by the curves , , and .
2. The region is revolved around the vertical line .
STEP 2
1. Set up the integral for the volume using the method of cylindrical shells.
2. Determine the limits of integration.
3. Evaluate the integral to find the volume.
STEP 3
Set up the integral for the volume using the method of cylindrical shells. The formula for the volume using cylindrical shells is:
where is the radius of the shell and is the height of the shell.
For this problem:
- The radius is the distance from the line to the curve , which is .
- The height is the function .
STEP 4
Determine the limits of integration. The region is bounded by and .
Thus, the limits of integration are from to .
STEP 5
Evaluate the integral to find the volume:
Simplify and evaluate the integral:
Evaluate each integral separately:
Substitute back and evaluate from 0 to 9:
Calculate:
Thus, the volume of the solid is:
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