Using the Washer Method to Calculate Volumes of Solids of Revolution | A Comprehensive Guide with Examples

washer method

The washer method is a technique used in calculus to calculate the volume of a solid of revolution

The washer method is a technique used in calculus to calculate the volume of a solid of revolution. Specifically, it is used when the solid is formed by rotating a region between two curves around a given axis.

To understand the concept, let’s consider a simple example. Suppose we have a region bounded by two curves, f(x) and g(x), on the interval [a, b]. To find the volume of the solid formed by revolving this region around the x-axis, we can use the washer method.

The first step is to take a vertical slice of the solid perpendicular to the x-axis, which creates a thin washer-shaped solid. The thickness of this washer is represented by delta x, where x is the variable representing the position along the x-axis.

The volume of this washer can be calculated by subtracting the inner radius (r1) from the outer radius (r2) and multiplying it by the height (h) of the washer. The inner radius is determined by the distance between the curve g(x) and the axis of rotation (in this case, the x-axis), while the outer radius is determined by the distance between the curve f(x) and the axis of rotation.

The formula for calculating the volume of a washer is as follows:

V = ∫[a, b] π[(r2)^2 – (r1)^2] dx

Where π is the mathematical constant pi, r1 is the inner radius, r2 is the outer radius, and dx represents the infinitesimally small change in x.

To find the inner and outer radii, we use the equations:

r1 = g(x)
r2 = f(x)

By integrating the formula from a to b, we can find the total volume of the solid generated by rotating the region between the curves f(x) and g(x) around the x-axis.

It’s worth noting that the washer method can also be used to find volumes of solids of revolution around other axes, such as the y-axis, by adjusting the equations accordingly.

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