QuestionCalculate the volume of the solid formed by rotating the area between , , , and around the -axis.
Studdy Solution
STEP 1
Assumptions1. The region is enclosed by the lines , , , and .
. The region is rotated about the -axis to form a solid.
3. The volume of the solid is found using the disk method.
STEP 2
The disk method formula is given bywhere is the volume of the solid, is the function defining the region, and is the interval over which the region extends.
STEP 3
In this problem, and the interval is .
STEP 4
Substitute and the interval into the disk method formula.
STEP 5
implify the integrand.
STEP 6
Now, we can calculate the integral.
STEP 7
Evaluate the integral at the upper and lower limits of the interval.
STEP 8
implify the expression.
STEP 9
Calculate the volume.
STEP 10
The volume of the solid formed by rotating the region enclosed by the given lines about the -axis is approximately cubic units.
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