Shell Method
int(a,b) (2pirh)
The shell method is a technique for finding the volume of a solid of revolution when the axis of revolution is parallel to the y-axis, and the function is rotated about the y-axis. It is named so because thin cylindrical shells are used to calculate the volume.
Here are the steps to use the shell method:
1. Visualize the solid of revolution that you want to find the volume of. Be sure that its axis of rotation is parallel to the y-axis.
2. Identify the lower and upper limits of integration. These are the values of x where the solid intersects the axis of rotation.
3. Set up an integral for the volume of one of the cylindrical shells. This can be determined using the formula 2πrh, where r is the distance from the axis of rotation to the outer edge of the shell, and h is the thickness of the shell.
4. Determine the limits for the radius of the shell, which will correspond to the lower and upper bounds of integration. These represent the distances between the axis of rotation and the edge of the solid.
5. Evaluate the integral over the given bounds in order to find the volume of the solid.
Tips for using the shell method:
1. Make sure that you have correctly identified the function that you are rotating.
2. Choose the correct variable to integrate with. In the shell method, this will be x.
3. Consider breaking up the solid into multiple sections if necessary, in order to simplify the integral.
4. When determining the thickness of the shell, consider taking the limit as the thickness approaches zero, in order to ensure an accurate result.
5. Check your work by visualizing the solid of revolution and comparing the calculated volume to your expectations.
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