Question
Calculate the volume of the solid generated by revolving the region bounded by the curve
and the line
around the x-axis.
Ask by Wheeler Owen. in Netherlands
Jan 05,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The volume of the solid is
.
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To find the volume of the solid generated by revolving the region bounded by the curve
and the line
around the x-axis, we can use the disk method. First, we determine the points of intersection of the curve and the line.
Setting
equal to
:
Thus, the solutions are
and
.
The volume
of the solid of revolution is given by the integral:
where
is the distance from the x-axis to the curve, which in this case would be the line
minus
.
Thus,
.
Now, we plug this into our volume formula:
Expanding the integrand:
So we need to compute:
We’ll evaluate the integral term by term:
-
- Since
is an even function,
- For
,
Combining these results, we find:
Finding a common denominator (which is 15):
So,
Thus, the final volume
is: