Direct applications and motivation of trig substitution for beginning calculus students

Motivating what is often called "Calculus 2" can be hard, which is probably why there are multiple other attempts at motivating it here. I have just begun teaching such a course, beginning with the bag-of-tricks for performing integration.

We will be spending a large amount of time on trig substitution, by which I mean evaluating integrals like

$$\int \frac{1}{\sqrt{a^2 - x^2}} \mathrm d x$$

by substituting $x = a \sin \theta$, and similar integrals of that type. This was similarly emphasized when I first learned calculus years ago, but though I can defend many aspects of a "calculus 2" course in terms of use or intellectual interest, I don't have a repository of interesting or motivating reasons why we spend so much time on this subject.

In principle, these types of problems might come out of physical phenomena, as these are very Pythagoreanesque.

So I wonder:

What are specific applications and motivations for learning trig substitution?

I have started with some simple applications - but I'm really hoping for more.

• "I don't have a repository of motivating reasons why we spend so much time on this subject" -- agreed. I look forward to the day when that time is much reduced. – user173 Sep 4 '14 at 2:16
• Apparently, some of the historical interest in these integrals is tied to the development of classical algebraic geometry. The $t=\tan(x/2)$ substitution has application to find the rational points on a circle. See pages 5 and 6 of amazon.com/Algebraic-Geometry-An-Introduction-Universitext/dp/…. Beyond interesting history, all you need is a quadratic in a model and a derivative and there it is, to solve it you integral a root. Beyond this, these are far more interesting when taught in parallel with hyperbolic substitutions. See section 9.3 of... – James S. Cook Sep 4 '14 at 3:32
• of supermath.info/OldschoolCalculusII.pdf where I compare and contrast some integrals from the trigonometric verses hyperbolic technique. Btw, variational calculus is full of differential equations whose explicit solution rest on the solution of integrals with roots. – James S. Cook Sep 4 '14 at 3:34

In my opinion, trig substitution is presented in a terrible fashion in every calculus book I have ever seen. "If you see $\sqrt{a^2 - x^2}$, substitute $x = a \sin \theta$, and then use such-and-such trig identity, blah, blah, blah..." Yet another unmotivated rule to memorize.

I always present trig substitution as follows: If you see any algebraic expression that looks like the Pythagorean theorem (i.e., $\sqrt{a^2 - x^2}$, $\sqrt{x^2 - a^2}$, or $\sqrt{x^2 + a^2}$), then DRAW A TRIANGLE such that the algebraic expression is one of the side lengths, choose an angle $\theta$, use the triangle to write out the values of trigonometric functions involving $\theta$, and substitute.

This takes three unmotivated "rules" off the table, and gives the students a different rule in their place: when in doubt, appeal to geometry.

• And I used to add: "Never, never, try to memorize the 'correct' substitution". – Bernard Masse Sep 4 '14 at 20:06
• As a successful undergrad student, grad student, and now professor, I have never found a time when I would have been better off having memorized the right substitutions. – rnrstopstraffic Jan 30 '18 at 5:32

There are some areas that are naturally calculated with trig substitution, and which appear somewhat naturally. For example,

1. Finding the area of the ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ can be done with trig substitution.
2. Given two intersecting circles in the plane, finding the area enclosed within one circle but outside the other can be done with trig substitution. This shape is a lune.
3. Finding the area within a circle and under a specified height leads to a trig sub integral.
4. Equivalently, finding the amount of soda in a cylindrical can lying on its side given the height of the soda leads to a trig sub.
5. A torus can be constructed by rotating a circle around the origin as a volume of revolution. Finding the area of this torus leads to a trig sub.

In general, finding areas of regions bounded by quadratic algebraic curves might often lead to integrals that can be tackled with trig substitution.

• Regarding #5: yes, finding the volume of a torus via integration does lead to a trigonometric substitution. However, it's much easier to recognize the torus as a cylinder wrapped around and ajoined at its circular bases. So, its volume is really based on the $\pi r^2\cdot h$ formula for a cylinder! – Brendan W. Sullivan Oct 17 at 13:18

Although it's not particularly specific, given a rational function $F(\sin(x), \cos(x))$, the substitution $z = \tan\frac{x}{2}$ will transform the integral $\displaystyle \int F(\sin x, \cos x) \mathrm d x$ into an integral of a rational function of $z$, which can in turn (and in principle) always be integrated with partial fractions.

These come up when calculating the average angular velocity of the output shaft of a universal joint when the two shafts are not aligned (as in the picture on wikipedia).

Unfortunately, I do not have intuition for coming up with interesting questions yielding rational functions of $\sin x$ and $\cos x$ (even though I have actually used this before!), so I cannot list more.

One possible motivation originates in a very simple problem. Suppose a boat located at the origin is tied to a buoy located at $(0, l)$ (and where the rope starts out taut). If the boat travels upwards on the $y$-axis, the buoy will travel in a path described by a function $f(x)$.

Since the buoy is always pulled in the direction of the boat, the line from the boat to the buoy is a tangent to $f(x)$, and so we can write down the differential equation

$$f'(x) = \frac{-\sqrt{l^2 - x^2}}{x},$$

which is clearly separable and clearly leads to a classical trig substitution problem. $\diamondsuit$

Similarly, any object pulling other free-moving objects with rigid materials of fixed length lead to isomorphic questions.

• The initial buoy location should be on the $x$-axis, at $(l,0)$. – awwalker Sep 26 '18 at 22:11

Although beyond the level of most students the first time they're learning trig substitution, a variety of problems asking about the electric field around uniformly charged 1-dimensional lines in space end up yielding integrals that should be done with trig substitution.

For instance, a uniformly charged rod of length $L$ lying with one end at the origin and the other at the point $(0,l)$ produces an electric field at the point $(x,y)$ given by

$$\int_{-x}^{l-x} \frac{y}{4\pi \epsilon_0(t^2 + y^2)^{3/2}} \mathrm{d}t.$$

Trigonometric substitution is related to how Newton first showed the derivative of $\sin x$ is $\cos x$. See https://hsm.stackexchange.com/questions/3174/how-were-derivatives-of-trigonometric-functions-first-discovered?rq=1

1. Agree with the draw a triangle advice from Thorne. This is actually not so uncommon in books, though--just looked at a few. Maybe, a good pedagogy is to push it with students though, so they don't have as much memorizing rules.

2. Looked through a few books and most of the problems were just straight symbols are rather artificial word problems (volumes of revolution for given shapes).

3. One real area that they are useful is in hydraulics, especially for open channels (not filled pipes). Granville book has a short chapter on various hydraulics problems. Also Paul's online notes give an example.

Paul's: http://tutorial.math.lamar.edu/Classes/CalcII/HydrostaticPressure.aspx (towards bottom)

Granville: Sorry,no link. The free pdf doesn't have the same chapter...grr. My copy is 1944 edition.

There are several electrostatics problems requiring trig substitution. (Google search will show this, but I have a copy of Wangness in hand.) And electrostatics as taught in physics classes, not design work using FE, does a lot of analytical solutions in problems. Physicists like students to be able to work out some of the problems too--not just hand it to Mathematica. I would suspect that various applied problems in other engineering courses also occasionally have an integral needing trig substitution.

All that said, my advice would not be to teach this by physical examples. It will probably just make it harder. Treat it as another trick in the bag of tricks. If there are questions about application, you can cite some of these (read up on them a little bit so you are not just parroting something I say, but know it yourself Or just tell the kids, that there are a lot of integrals coming at them in physics and engineering courses and they need to have a decent toolkit.

[I think exposure to some decent variety of tougher integration methods, partial fractions and all that, makes students more confident in using integral tables also...at least they know they have seen a lot of stuff and worked with a lot of stuff. Not just blindly consulting the CRC. But this would be for kids who are taking a solid STEM course. Not a reduced calc class.]

Honest, I think part of the benefit of learning some things in math class is getting some of the learning in a more abstract fashion (with x's and y's) instead of physical variables. Let's the student learn it in that context. Then when they see it in physical science course or engineering, they learn it that way also.

Can be good to see things a couple different ways. Even if it is a year later, when they randomly see a trig substituation in a derivation or homework problem of a sci/eng class and they have forgotten it, they will remember they learned it once. So the discussion in sci/eng class will be easier than if this was first time. And the subsequent exposure may help solidify the concept or at least briefly refresh it and leave the student confident of knowing he could look up the method in his text or in a table of integrals.