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How do you use cylindrical shells to find the volume of a solid of revolution?

Answer 1

If the solid is obtained by rotating the region between the graph of #f(x)# and the x-axis from #x=a# to #x=b# about the y-axis, the volume #V# of the solid can be found by Shell Method: #V=2pi int_a^b xf(x) dx#

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Answer 2

Emily Ammerman

To find the volume of a solid of revolution using cylindrical shells, follow these steps:

Determine the axis of revolution.
Determine the limits of integration along the axis of revolution.
Slice the solid into thin cylindrical shells perpendicular to the axis of revolution.
Express the volume of each shell as the product of its height, circumference, and thickness.
Integrate the volumes of all the shells over the appropriate interval along the axis of revolution using the shell method formula: V = ∫(2πrh)dx, where r is the distance from the axis of revolution to the shell, h is the height of the shell, and dx represents the differential distance along the axis of revolution.

By integrating the volumes of all the shells, you will find the total volume of the solid of revolution.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.