We can determine the volume of a solid with similar cross-sections using definite integrals. In this post, let's explore the washer method in finding the volume of a solid of revolution.
We move on to a particular case of finding volume \(V\) using integration. For these cases, we will be dealing with a solid of revolution. There are three general variations of finding \(V\).
This post discusses the second of these: the washer method.
Approximation
Let's say you're finding the volume \(V\) of a solid of revolution with a hollow core by approximating it first. The object consists of a uniform cross-section throughout its whole length. We generated the solid by revolving the area along its axis of rotation, which doesn't lie along the area boundary.
We must divide the solid into multiple parts to estimate the total volume. We can imagine these parts as a series of similar plane sections stacked on each other to make up the 3D object.
For a solid of revolution, we can divide it into either discs or shells. For this post, we will use discs.
There is a problem when we use discs in this example - we have an open portion in the middle. Rather than using a solid disc, we use a hollow version of it which we call washers.
From this, we begin the process of approximating:
First, divide the 3D object into multiple regular washers.
Then, we solve the volume \(V\) of each divided washer. From Cavalieri's theorem, \(V=Ah\) where \(A\) is the uniform cross-sectional area and \(h\) is the length of the solid. Since it is a circular washer, \(A\) is equal to \(\pi (R^2-r^2)\) where \(R\) is the outer radius and \(r\) is the inner radius.
Finally, we add all individual \(V\) to get the total approximate volume.
Since this is an approximation, how can we have a more accurate answer? We increase the number of washers \(n\).
When we increase the number of washers, it will be closer to the actual value.
As the number of washers increases, the smaller its volume \(dV\) and thickness \(h\) become.
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From the approximate solution, we know that if we want an exact answer, the volume and height of the washers must be tiny. Building from this, we can solve the precise volume. We will only need to modify two things in the approximate solution:
We need to set up a solid washer that is so small we can express its volume as the differential \(dV\)
We need an operation that will sum up all of these tiny divisions
We can combine all \(dV\) through integration; hence, the general equation to solve for the volume \(V_T\) is:
\(V_T=\int_{a}^{b}{dV}\)
\(V_T\) is the total volume
\(dV\) is the differential volume of the division.
\(a\) is the lower limit position (endpoint with the lowest value)
\(b\) is the upper limit position (endpoint with the highest value)
This equation is similar to solving the volume of solids using integration with uniform cross-sections. What makes this unique is the differential partition which is always a washer; hence, the name: the washer method.
One essential detail for differential washers (or discs) is that they must be perpendicular to the axis of rotation.
Differential Washer
Let's discuss the differential washer with volume \(dV\). Using Cavalieri's theorem, the differential volume is \(dV=A \cdot dh\).
\(A\) is the cross-sectional area of the solid
\(dh\) is the differential thickness
The cross-sectional area is a hollow circle. We can further expand this as: \(A=\pi (R^2-r^2)\); hence, \(dV\) becomes:
\(dV=\pi (R^2-r^2)dh\)
Let's expound more on these variables:
The variables \(R\) and \(r\) are the outer and inner radius functions, respectively. We need to express these in terms of the differential variable. The radius function will depend on the orientation of the solid if it is along the x, y, or z-axis.
The differential thickness \(dh\) is the distance between two points; however, because it is so minuscule, we represent it with a position variable \(x\), \(y\), or \(z\). Again, the thickness will also depend on the orientation.
These variables \(R\), \(r\), and \(dh\) must be consistent with the differential variable.
Going back to \(V_T\), we can summarize the washer method with the following general equation:
\(V_T=\pi \int_{a}^{b} (R^2-r^2) dh\)
This method is a variation of the disc method. It deals with cases where the disc has a hollow portion in the middle.
Summary
Solving the volume by integration expands on the approximation solution.
The logic is to subdivide the volume needed with multiple solids, solve for the volume of each part, and add all of these volumes.
It is necessary to keep the volume small to increase the accuracy of the solution; hence, we must represent the volume as differential \(dV\).
We can combine all \(dV\) through integration; hence, the general equation to solve for the volume is: \(V_T=\int_{a}^{b}{dV}\).\(V_T\) is the total volume, \(dV\) is the differential volume of the division, \(a\) is the lower limit position (endpoint with the lowest value), and \(b\) is the upper limit position (endpoint with the highest value)
We can expound the differential solid with volume \(dV\) with Cavalieri's theorem: \(dV=A \cdot dh\). \(A\) is the cross-sectional area of the solid, and \(dh\) is the differential thickness.
The cross-sectional area is always a hollow circle since we deal with a washer. We can further expand the area as: \(A=\pi (R^2-r^2)\); hence, \(dV\) becomes: \(dV=\pi (R^2-r^2)dh\)
One essential detail for differential washers (or discs) is that they must be perpendicular to the axis of rotation.
The variables \(R\) and \(r\) are the outer and inner radius functions, respectively. These functions will depend on the orientation of the solid if it is along the x, y, or z-axis.
The differential thickness \(dh\) is the distance between two points; however, because it is so minuscule, we represent it with a position variable \(x\), \(y\), or \(z\) which will depend on the orientation.
These variables \(R\), \(r\), and \(dh\) must be consistent with the differential variable.
We can summarize the washer method with the following general equation: \(V_T=\pi \int_{a}^{b} (R^2-r^2) dh\)
This method is a variation of the disc method. It deals with cases where the disc has a hollow portion in the middle.
