How do you use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by #y = 32  x^2# and #y= x^2# 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 = 32  x^2 ) and ( y = x^2 ) revolved about ( x = 4 ), follow these steps:

Determine the limits of integration by finding the points of intersection between the two curves: ( 32  x^2 = x^2 ) Solve for ( x ) to find the points of intersection.

Set up the integral for the volume using the formula for cylindrical shells: [ V = \int_{a}^{b} 2\pi x f(x) dx ] where ( f(x) ) represents the height of the shell and ( x ) is the distance from the axis of rotation (in this case, ( x = 4 )).

Calculate the height of the shell, which is the difference between the upper and lower curves at a given ( x ) value.

Integrate the expression from step 2 over the interval determined in step 1 to find the volume of the solid of revolution.
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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.
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