Is This Calculation for the Volume of a Solid of Revolution Correct?

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SUMMARY

The calculation for the volume of a solid of revolution about the y-axis, using the method of cylindrical shells, is confirmed to be correct. The volume is derived from the integral of the area function A(x) = 2π(radius)(height), where the radius is x and the height is the difference between the two functions: x^(1/5) - (2x^2 - 3x + 2). The integration from x = 0.619 to x = 1 yields a volume of approximately 0.169π, which aligns with the output from Maple, confirming the accuracy of the setup.

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  • Understanding of calculus, specifically integration techniques
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  • Knowledge of graphing functions to find intersection points
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  • Study the method of cylindrical shells in detail for volume calculations
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stephencormac
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Can anyone confirm if I have done the following work correctly

Find the volume of a solid of revolution obtained by rotating about the y-axis the region bounded by y = the fifth root of x and 2x^2 - 3x + 2.

By drawing the graph, I figured out that I need to use the method of cylindrcal shells given by v = integral from 0.619 to 1 of A(x) dx.

Where A(x) = 2pi(radius)(height)

The intersection points of the equations are approximately x = 1 and x = 0.619

Radius is equal to x.

Height is equal to the difference in the two equations

i.e. (x^(1/5) - 2x^2 + 3x - 2)

Thus we have 2pi(radius)(height)

=2pi*(x)*(x^(1/5) - 2x^2 + 3x - 2)

= 2pi*(x^(6/5) - 2x^3 + 3x^2 - 2x)

Now I will integrate this between 0.619 and 1

= (5/11)x^(11/5) - (1/2)x^4 + x^3 - x^2 ¦ 0.619 to 1
which gives me 2pi*(-0.05 + 0.1345)
=0.169pi approximately.

Can anyone confirm that I have done this correctly
 
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Your setup looks OK. Maple gives V := .03140665150*Pi.
 

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