How Do You Calculate Volume by the Shell Method for Rotated Solids?

[whatlifeforme]

Homework Statement

Find the volume of the solid generated by revolving the region bounded by y=4x-x^2 and y=x about the y-axis and about the line x=3.

Homework Equations

1. y=4x-x^2 and y=x
2. y=4x-x^2 and x=3

The Attempt at a Solution

1. 2∏∫ (0 to 3) (x)(4x-x^2 -x) dx
2. 2∏∫ (0 to 3) (3-x)(4x-x^2 -x) dx

 

[HallsofIvy]

Obviously you need to go ahead and actually subtract that last "x" in each formula: $2\pi\int_0^3 x(3x- x^2)dx= 2\pi\int_0^3 3x^2- x^3 dx$ and $2\pi\int_0^3 (3- x)(3x- x^2)dx= 2\pi\int_0^3 9x- 6x^3+ x^3 dx$ but, yes, those are set up correctly.

 

[Mark44]

Homework Statement

Find the volume of the solid generated by revolving the region bounded by y=4x-x^2 and y=x about the y-axis and about the line x=3.

Homework Equations

1. y=4x-x^2 and y=x
2. y=4x-x^2 and x=3

The Attempt at a Solution

1. 2∏∫ (0 to 3) (x)(4x-x^2 -x) dx
2. 2∏∫ (0 to 3) (3-x)(4x-x^2 -x) dx

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