# How do you find the volume of a solid that is enclosed by #y=3x^2# and y=2x+1 revolved about the x axis?

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To find the volume of the solid enclosed by ( y = 3x^2 ) and ( y = 2x + 1 ) revolved about the x-axis, you can use the method of cylindrical shells. Follow these steps:

- Determine the limits of integration by finding the points of intersection between the two curves ( y = 3x^2 ) and ( y = 2x + 1 ).
- 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 at each x-value, and ( a ) and ( b ) are the limits of integration.
- Calculate the height of each cylindrical shell at a given x-value by finding the difference between the y-values of the upper and lower curves.
- Integrate the expression from step 2 with respect to ( x ) over the determined limits of integration.
- Evaluate the integral to find the volume of the solid.

These steps will give you the volume of the solid formed by revolving the region between the curves ( y = 3x^2 ) and ( y = 2x + 1 ) about the x-axis.

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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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