How do you use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by #y = x^3#, y=0 , x=1 revolved about the y=1?
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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 = x^3 ), ( y = 0 ), and ( x = 1 ) revolved about ( y = 1 ), follow these steps:

Determine the limits of integration. In this case, the region is bounded by ( y = x^3 ), ( y = 0 ), and ( x = 1 ), so the limits of integration for ( y ) will be from 0 to 1.

Express the radius (( r )) and height (( h )) of the cylindrical shell in terms of ( y ). The radius is the distance from the axis of rotation (( y = 1 )) to the curve ( y = x^3 ), which is ( 1  \sqrt[3]{y} ). The height of the shell is the difference in ( x )values, which is ( 1  0 = 1 ).

Set up the integral for the volume using the formula ( V = \int_{a}^{b} 2\pi rh , dy ), where ( a ) and ( b ) are the limits of integration for ( y ).

Integrate the expression ( 2\pi rh ) with respect to ( y ) from the lower limit to the upper limit.

Solve the integral to find the volume of the solid.
Following these steps, the volume of the solid obtained by rotating the region about ( y = 1 ) can be calculated using the method of cylindrical shells.
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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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