How to Solve the Volume Using the Shell Method?

Exact Solution

From the approximate solution, we know that if we want an exact answer, the volume and thickness of the shells 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 shell 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 shell; hence, the name: the shell method.

One essential detail for differential shells is that they must be parallel to the axis of rotation. 

Differential Shell

Let's discuss the differential shell with volume \(dV\). From our derivation of the shell's volume, we can say that \(dV\) is:

\(dV=2 \pi R h \cdot dw\)

Let's expound more on these variables:

• The radius and height functions are the variables \(R\) and \(h\). These functions will depend on the orientation of the solid if it is along the x, y, or z-axis. We need to express these in terms of the differential variable.
• The differential thickness \(dw\) 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 \(R\), \(h\), and \(dw\) must be consistent with the differential variable.

Going back to \(V_T\), we can summarize the shell method with the following general equation:

\(V_T=2\pi \int_{a}^{b} R h \cdot dw\)

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)

The differential solid is a shell in which the volume \(dV\) is \(dV=2 \pi R h \cdot dw\)

One essential detail for differential shells is that they must be parallel to the axis of rotation.

The variables \(R\) and \(h\) are the radius and height functions of the shell, respectively. These functions will depend on the orientation of the solid if it is along the x, y, or z-axis.

The differential thickness \(dw\) 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 \(R\), \(h\), and \(dw\) must be consistent with the differential variable.

We can summarize the shell method with the following general equation: \(V_T=2\pi \int_{a}^{b} R h \cdot dw\)