How do you use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region #y=6x^2#, #y=6sqrtx# rotated about the yaxis?
The normal revolution method calls for:
In contrast, the shell method calls for a volume formula as such:
Let's see how this looks.
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To use the shell method for finding the volume of the solid generated by revolving the region ( y = 6x^2 ) and ( y = 6\sqrt{x} ) about the ( y )axis, follow these steps:

Determine the limits of integration by finding the intersection points of the curves ( y = 6x^2 ) and ( y = 6\sqrt{x} ).

Express the differential volume element ( dV ) in terms of ( y ) and ( dy ).

The integral to find the volume using the shell method is: [ V = 2\pi \int_{a}^{b} y \cdot \text{radius} \cdot dy ] where ( a ) and ( b ) are the lower and upper limits of integration respectively, ( y ) is the variable of integration, and the radius is the distance from the axis of rotation to the outer edge of the solid.

The radius can be found by expressing ( x ) in terms of ( y ) and finding the distance from the ( y )axis to the curve.

Evaluate the integral using the determined limits of integration and the appropriate expression for the radius.
Following these steps, the integral should give the volume of the solid generated by revolving the given region about the ( y )axis using the shell method.
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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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