# How do you find the volume of the solid generated by revolving the region bounded by the curves #y=2x^2 -x^3# and y = 0 rotated about the y-axis?

This is what you will be revolving.

graph{2x^2 - x^3 [-0.1, 10, -0.3, 5]}

What we could do here is use the shell method, which is far more convenient in this case than the traditional revolution method.

where:

Now where is the x-intercept? We can see it's

So we have:

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To find the volume of the solid generated by revolving the region bounded by the curves ( y = 2x^2 - x^3 ) and ( y = 0 ) rotated about the y-axis, we can use the method of cylindrical shells.

- Determine the limits of integration. Set the two equations equal to each other and solve for ( x ) to find the points of intersection.

[ 2x^2 - x^3 = 0 ]

[ x(2 - x^2) = 0 ]

[ x = 0, \text{ and } x = \pm \sqrt{2} ]

So, the limits of integration are ( x = 0 ) to ( x = \sqrt{2} ).

- Set up the integral for the volume using the formula for the volume of a cylindrical shell:

[ V = 2\pi \int_{a}^{b} x \cdot h(x) , dx ]

Where ( h(x) ) is the height of the shell at the given value of ( x ), and ( a ) and ( b ) are the limits of integration.

In this case, ( h(x) ) represents the distance between the curves at each ( x )-value. So, ( h(x) = 2x^2 - x^3 ).

- Integrate from ( x = 0 ) to ( x = \sqrt{2} ):

[ V = 2\pi \int_{0}^{\sqrt{2}} x \cdot (2x^2 - x^3) , dx ]

- Evaluate the integral.

[ V = 2\pi \int_{0}^{\sqrt{2}} (2x^3 - x^4) , dx ]

[ V = 2\pi \left[ \frac{1}{2} x^4 - \frac{1}{5} x^5 \right]_{0}^{\sqrt{2}} ]

[ V = 2\pi \left( \frac{1}{2} \cdot (\sqrt{2})^4 - \frac{1}{5} \cdot (\sqrt{2})^5 - \frac{1}{2} \cdot 0^4 + \frac{1}{5} \cdot 0^5 \right) ]

[ V = 2\pi \left( \frac{2}{2} - \frac{2\sqrt{2}}{5} \right) ]

[ V = 2\pi \left( 1 - \frac{2\sqrt{2}}{5} \right) ]

[ V = 2\pi - \frac{4\pi\sqrt{2}}{5} ]

[ V = \frac{10\pi}{5} - \frac{4\pi\sqrt{2}}{5} ]

[ V = \frac{10\pi - 4\pi\sqrt{2}}{5} ]

So, the volume of the solid generated is ( \frac{10\pi - 4\pi\sqrt{2}}{5} ).

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When evaluating a one-sided 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 one-sided 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 one-sided 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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