Calculating Volume With The Washer Method: A Mathematical Guide To Solids Of Revolution

Washer Method

int(a, b) (piR^2-pir^2)

The washer method is a mathematical procedure used to find the volume of a solid of revolution, when it is rotated around an axis that lies outside the object.

To use the washer method, you first need to have a 2D shape, typically a curve that is shown in a coordinate plane. This curve may or may not be a function, but it should be bounded by two vertical lines for the region of interest, say the interval [a,b]. Let’s assume this curve is the graph of a function, f(x), over the interval [a,b]. Now, we can represent the curve as a collection of horizontal or vertical slices as shown below:


To find the volume using the washer method, we need to:

1. Split the region of interest, [a,b], into n equal sub-intervals.
2. Choose a representative value, xi, in each sub-interval i, such that a <= xi <= b. 3. For each sub-interval, calculate the area A of the outer circle by using the formula, A = π(Ri^2), where Ri is the distance from the axis of rotation to the outer edge of the solid, and is given by Ri = f(xi)+R, where R is the distance from the axis of rotation to the inner edge of the solid. 4. Similarly, calculate the area a of the inner circle by using the formula, a = π(r_i^2), where r_i is the distance from the axis of rotation to the inner edge of the solid, and is given by r_i = f(xi). 5. The volume of the solid can then be obtained by adding up the volumes of the cylindrical shells formed by subtracting the inner circle areas from the outer circle areas, over the whole region of interest, that is: V = ∫[a,b] π[Ri^2 - r_i^2] dx where V is the volume of the solid and dx is an infinitesimal length element of each washer. In summary, the washer method is a geometric technique used to find the volume of a solid of revolution that is formed by rotating a 2D curve around an axis that is outside the object of interest. The method involves slicing the cone into thin, cylindrical washers or shells with the thickness of the shell being an infinitesimal dx.

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