# How do you find the volume of the solid generated by revolving the graph of a function #f(x)# around a point on the x-axis?

point.

This formula is based on

given range for x.

Here, an elementary area , in the form of a rectangle of length

to generate an elementary solid of revolution that is in the form of a

volume for summation is

Then, it is summation of the infinite series for V, in the limit.

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To find the volume of the solid generated by revolving the graph of a function ( f(x) ) around a point on the x-axis, you can use the disk method or the washer method.

Disk Method:

- Determine the axis of rotation and the limits of integration.
- Slice the solid into thin disks perpendicular to the axis of rotation.
- Express the volume of each disk as ( V = \pi (f(x))^2 \Delta x ).
- Integrate the expression ( \pi (f(x))^2 ) with respect to ( x ) over the given bounds of integration.

Washer Method:

- Determine the axis of rotation and the limits of integration.
- Slice the solid into thin washers perpendicular to the axis of rotation.
- Express the volume of each washer as ( V = \pi ((\text{outer radius})^2 - (\text{inner radius})^2) \Delta x ).
- Integrate the expression ( \pi ((\text{outer radius})^2 - (\text{inner radius})^2) ) with respect to ( x ) over the given bounds of integration.

After setting up the integral using either the disk method or the washer method, evaluate the integral to find the volume of the solid.

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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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