Washer Method
int(a, b) (piR^2-pir^2)
The Washer Method is a technique used in calculus to calculate the volume of an object generated by the rotation of a function around a given axis. This method involves dividing the object into thin, concentric slices, or washers, and integrating their individual volumes to determine the total volume.
The formula for the Washer Method can be derived by following these steps:
1. Determine the axis of rotation and the bounds of integration.
2. Find the outer and inner radii of a typical washer by subtracting the function values at the two relevant points on the axis of rotation.
3. Calculate the area of the washer by subtracting the area of the inner circle from the area of the outer circle: A = π(R² – r²).
4. Multiply the area of the washer by its thickness (dx or dy, depending on the axis of rotation) to get its volume: dV = A(dx or dy).
5. Integrate these individual volumes over the given bounds of integration to find the total volume of the object.
It’s important to note that the Washer Method is used for revolving around a horizontal or vertical line, rather than an oblique or slanted axis. Additionally, the function being rotated must be continuous and non-negative over the defined bounds of integration.
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