Professor Leonard – Calculus 3
14.4 – Center of Mass (and Moments of Mass and Inertia) for Lamina in 2-D
14.6 – How to Solve Triple Integrals (Along with Center of Mass and Volume)
Thomas – 15.5 Triple Integrals in Rectangular Coordinates; 15.6 Moments & Centers of Mass
We’ve come a long way. In Calculus we first learn that integrals are the counterpart to derivatives. The single integrals that we first work with
- \int_a^b f(x)\,dx
Can be used to calculate area under a 2D function or the mass of a straight line.
In Calculus 3 we first encounter double integrals.
- \iint_R f(x,y)\,dA
The addition of a second variable serves to calculate the volume under a surface or the mass of an infinitesimally thin plate.
No sooner do we start to wrangle double integrals than we throw in a third variable
- \iiint_T f(x,y,z)\,dV
From the first two integrals, we see that we’re returned an additional dimension, e.g. f’(x) → (x,y) , f’(x, y) → (x, y, z) , or the mass of geometric structure with the same number of dimensions as variables. As such, a triple integral can calculate some four-dimensional space under/within a 3D space (however that works). Or, more practically, it allows us to calculate the mass of a 3D region.
Integration with three variables is ultimately much like integration of two variables. We first choose which axis we’ll use to draw our limits of integration, and then we’ll define said limits with the remaining two variables. Once we’ve chosen our R3 axis to start with, integration follows the same rules as with double integration.
We learned previously from Fubini’s Theorem that the order of double integration can be swapped without changing our calculation. Similarly, the order of triple integration is only consequential insofar as it affects how easy or difficult it is to perform the integration. The end result is the same.
This is really cool, as I can ‘see’ actual usage of this as it can directly apply to the three-dimensional world in which I inhabit.
We apply these double and triple integrals to measurements of centers of mass, first moments of mass, and second moments of inertia. Centers of Mass (COM) are points, where the weighted relative position of all mass in a system sums to zero. These are like balance points. To find COM, we first calculate mass like this:
We start with mass functions. For a 2D mass over a lamina R:
- M=\iint_R \rho(x,y)\,dA
And for a 3D mass over a solid T:
- M=\iiint_T \rho(x,y,z)\,dV
We then measure our First Moments of Mass, which is the tendency of a lamina or object to rotate about an axis. A quick look at the integral makes sense of it all, as it’s simply the integral of the product of the axis and the mass density function.
Our First Moments of Mass in 2D are:
About the x-axis:
- M_x=\iint_R y\rho(x,y)\,dA
And about the y-axis:
- M_y=\iint_R x\rho(x,y)\,dA
The First Moments of Mass in 3D are:
About the yz-plane:
- M_{yz}=\iiint_T x\rho(x,y,z)\,dV
About the xz-plane:
- M_{xz}=\iiint_T y\rho(x,y,z)\,dV
About the xy-plane:
- M_{xy}=\iiint_T z\rho(x,y,z)\,dV
With these in hand, we can calculate our COM:
For a 2D lamina:
- \bar{x}=\frac{M_y}{M},\qquad \bar{y}=\frac{M_x}{M}
And for 3D bodies:
- \bar{x}=\frac{M_{yz}}{M},\qquad \bar{y}=\frac{M_{xz}}{M},\qquad \bar{z}=\frac{M_{xy}}{M}
Additionally, we can look at the Second Moments of Inertia, which is essentially how hard it is to get a lamina or object moving around an axis. We calculate them like this:
For 2D moments of inertia of a lamina:
About the x-axis:
- I_x=\iint_R y^2\rho(x,y)\,dA
About the y-axis:
- I_y=\iint_R x^2\rho(x,y)\,dA
About the origin (polar moment):
- I_0=\iint_R (x^2+y^2)\rho(x,y)\,dA
Notice the pattern compared with first moments: distance becomes distance squared.
For 3D moments of inertia of a solid T:
About the x-axis:
- I_x=\iiint_T (y^2+z^2)\rho(x,y,z)\,dV
About the y-axis:
- I_y=\iiint_T (x^2+z^2)\rho(x,y,z)\,dV
About the z-axis:
- I_z=\iiint_T (x^2+y^2)\rho(x,y,z)\,dV
The nice pattern here is: square the two coordinates perpendicular to the axis and add them.
I have really enjoyed learning about this, as it’s actually intuitive, and insightful into common, everyday phenomena. I can see how this could be integral, no pun intended, to Physics and Engineering.
Working through these problems was challenging in two ways. Firstly, I need to understand how to define my regions and set up my bounds. This requires me to understand how different forces influence the phenomena we are measuring. It helps greatly draw these out as best as we can (thanks, Professor Leonard). As I’ve mentioned before, I’m not the most spatially inclined math fan in the 4D space, so this alone gives me a good mental workout.
Secondly, I’m prone to bookkeeping errors. Even when setting up my integrals properly, with long, nested antiderivatives of trig functions and exponents mixed up in my chicken scratch writing, I can end up with an incorrect answer here and there.
While I like practice doing some by hand, I’ve increasingly handed off the heavy lifting to Wolfram Alpha https://www.wolframalpha.com/ which is just a badass calculator that tears through problems whose bookkeeping might otherwise have me tearing my hair out. Hats off to the brilliant Stephen Wolfram for this incredible tech.