How to Solve Volume Using Integration?

Exact Solution

From the approximate solution, we know that if we want an exact answer, the volume and height of its divisions 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 division 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 \(V_T\) with constant cross-section 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 the general expression we use to solve the volume of a 3D object. We can use this as long as the solid obeys Cavalieri's theorem. One critical requirement is that the solid must have a uniform cross-section throughout its length.

We will see this equation a lot, especially in finding the volume of solids of revolutions.

Differential Solid

Let's move on and expound on the differential solid with volume \(dV\). We can express it as \(dV=A \cdot dh\)

• \(A\) is the cross-sectional area of the solid
• \(dh\) is the differential thickness

Let's expound more on these variables:

• The cross-sectional area is a function. We need to express it in terms of the differential variable. The area 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.

Most importantly, these two variables \(A\) and \(dh\) must be consistent with the differential variable.

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 a function. The area 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 \(z\) which will depend on the orientation.

These two variables \(A\) and \(dh\) must be consistent with the differential variable.