How do you use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by #y = 1/x^4#, y = 0, x = 1, x = 4 revolved about the x=4?
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To use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by ( y = \frac{1}{x^4} ), ( y = 0 ), ( x = 1 ), and ( x = 4 ) revolved about the line ( x = 4 ), follow these steps:

Determine Limits of Integration: The region of integration lies between ( x = 1 ) and ( x = 4 ).

Setup the Integral: The volume ( V ) is given by the integral ( V = \int_{1}^{4} 2\pi rh , dx ), where:
 ( r ) is the distance from the axis of rotation to the shell, which is ( 4 + x ) in this case.
 ( h ) is the height of the shell, which is ( \frac{1}{x^4} ).

Evaluate the Integral: Integrate ( 2\pi (4 + x) \frac{1}{x^4} , dx ) from ( x = 1 ) to ( x = 4 ).

Calculate the Volume: Evaluate the definite integral obtained in the previous step to find the volume of the solid.

Final Step: Make sure to express the volume in exact or approximate form, as required.
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When evaluating a onesided 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.
When evaluating a onesided 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.
When evaluating a onesided 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.
When evaluating a onesided 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.
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