How do you find the volume of the solid obtained by rotating the region bounded by the curves #y=x#, #x=0#, and #y=(x^2)6# rotated around the #y=3#?
The grey region is what we will be rotating around the horizontal line
The outer radius is
The inner radius is
Using the method of washers
Integrating
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To find the volume of the solid obtained by rotating the region bounded by the curves ( y = x ), ( x = 0 ), and ( y = x^2  6 ) around the line ( y = 3 ), you can use the method of cylindrical shells.
The formula for the volume of a solid generated by revolving the region bounded by ( f(x) ), ( g(x) ), and the lines ( x = a ) and ( x = b ) around the line ( y = c ) is:
[ V = 2\pi \int_a^b (x  c) \cdot f(x)  g(x) , dx ]
Here, ( a ) and ( b ) are the xvalues of the intersection points of ( y = x ) and ( y = x^2  6 ), which are the solutions of the equation ( x = x^2  6 ).

First, find the intersection points by solving ( x = x^2  6 ). ( x = x^2  6 ) ( x^2  x  6 = 0 ) ( (x  3)(x + 2) = 0 ) ( x = 3 ) or ( x = 2 )

Now, set up the integral: ( V = 2\pi \int_{2}^3 (x  3) \cdot (x)  (x^2  6) , dx )

Compute the absolute value: When ( x ) is in the range ([2, 0]), ( (x)  (x^2  6) = (x^2  x + 6)  x = x^2  2x + 6 ) When ( x ) is in the range ([0, 3]), ( (x)  (x^2  6) = (x)  (x^2  6) = 6  x^2 + x )

Evaluate the integral: [ V = 2\pi \left( \int_{2}^0 (x  3)(x^2  2x + 6) , dx + \int_0^3 (x  3)(6  x^2 + x) , dx \right) ]
After evaluating this integral, you will get the volume of the solid.
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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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