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I am trying to teach my calculus students to apply integration by thinking about what they are integrating rather than just applying formulas. Calculus books are full of formulas like "to find the volume of a solid of revolution obtained by rotating region A around axis B, write down integral C". I don't want my students to just memorize these formulas, but to understand where they come from and to be able to apply integration in contexts where these formulas don't necessarily apply. What are some good examples of applications of integration that don't fit into the textbook formulas?

Here's a simple example: find the area of a circle by dividing it into concentric rings and integrating the area $2\pi x\, dx$ of each ring from 0 to $r$. Of course you can do the same finding the volume of a sphere.

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Are you just looking for in terms of geometry or any interesting applications of integrals? – ruler501 Mar 29 '14 at 4:40
@JyrkiLahtonen I think, your comment (with more details) might be better off as an answer. – Markus Klein Mar 29 '14 at 7:36
Give the students an $I$-$t$-graph of e.g. a capacitor loading process, the formula $I=\dot Q$ and ask them for the quantum of electrical charge in the capacitor. – Toscho Mar 29 '14 at 11:25
I remember there was a question about how many people at a concert given the people allowed in at any given time on the AP test a while ago. – ruler501 Mar 29 '14 at 18:49
up vote 14 down vote accepted

An example that illustrates a slightly different integral is the following.

enter image description here

An observer at point $O$ is exposed to traffic noise from a long straight highway (the line $AP$). The task is to show that the total noise level at $O$ is inversely proportional to the distance $r=|OA|$ from the highway.

Use the following approximations/simplifications.

  • The traffic density is constant along the road (a kind of an average).
  • The level of "atomic noise intensity" at $O$ follows the inverse square law (this is probably a good approximation you can easily justify).
  • The total noise intensity is the sum of the infinitesimal/atomic noise intensities (i.e. the quantity measured in Watts per square meter is additive. You only switch to the logarithmic decibel scale afterwards if ever, as things are not additive on the decibel scale).

So if you place $x$-axis along the road, and $P$ is at $x$, then the noise generated along a segment of the road of length $dx$ around $P$ has (at a unit distance from $P$) intensity $K\,dx$ for some constant $K$ proportional to traffic density (= number of passing vehicles per hour). When we take the distance $|OP|=\sqrt{r^2+x^2}$ into account we get $$ dI=K\,\frac{dx}{r^2+x^2} $$ for the contribution of this atomic noise to the total observed noise level at point $O$. The total noise intensity is thus $$ I=\int dI=\int_{-\infty}^{\infty}K\,\frac{dx}{r^2+x^2}. $$

So you get to illustrate/discuss:

  • improper integrals
  • indefinite integral of a simple rational function
  • discuss what goes wrong when $r\to0+$ (the approximations fail, and you cannot be at distance zero from a continuous source of the noise)
  • point out that a similar (but also different) integral emerges when you the field generated by an electric current along a long straight wire

I have tried this both as an in class and a homework example. As a HW problem it did create some discussion as the physics majors protested at first (assuming that I was adding decibels together). I guess the non-physics majors may have been bored. You know your customers (and if somebody can add an example from chemistry I will be very grateful).

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This looks excellent to me, thanks! (But then I do tend to like physics examples best myself.) – Mike Shulman Mar 30 '14 at 20:38

Physics provides with a lot of examples, whose main disadvantage is that they might be considered too much physics by some, i.e., require too much thinking or knowledge about something other than mathematics. (Also engineers and physicists might rather consider them the usual applications of integration.) I kept the examples simple, and as a result they might still be solved without understanding by guessing what to integrate, but they can easily made more difficult to avoid this.

Lengthening of an elastic rope under its own weight

Q: An elastic rope of length $l$ has a weight per length of $ρ$ (e.g., $ρ=1\,\text{kg}/\text{m}$) – both if unstretched. If a total of mass $m$ is pulling on a piece of rope, it extends to $f(m)$ of its unburdened length (typically $f(m)=1+Em$). To what length does the rope extend under its own weight?

A: A infinitesimally short piece of rope of length $\mathrm{d}x$ located at point $x$ (measured from the bottom of the rope in the unstretched case) is burdened with a weight of $ρx$ and thus its burdened length is $f(ρx)\mathrm{d}x$. The burdened length of the entire rope thus is the integral $\int_0^l f(ρx) \mathrm{d}x$.

A difficulty lies in considering the rope unstretched for most purposes (but this can be given as a hint). To “spice up” this problem, one could attach weights to the ropes (at the end or other positions) or have the rope vary in thickness (i.e., make $ρ$ as well as $f$ depend on $l$).

Uniformly accelerated object

Q: An apple starts falling from a tree at $t=0$. Its speed $v(t)$ at time $t$ is $gt$ (with $g$ being the local gravity of Earth). How far has it fallen after time $T$?

A: The distance covered by the apple during an infinitesimally short time $\mathrm{d}t$ is $v(t)\mathrm{d}t$ and thus the total distance covered up to time $T$ is $\int_0^T v(t)\mathrm{d}t = \int_0^T at\mathrm{d}t = \tfrac{1}{2}aT^2$.

To “spice up” the problem, one could consider more complicated acceleration functions (e.g., considering air resistance), initial speeds and so on.

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I think most of my students have memorized that distance is the integral of velocity, that being a common example in the book, so I'm not sure that example will be what I want. – Mike Shulman Mar 30 '14 at 14:27
The first one looks good, but I'm confused about what it means to talk about a rope burdened with a mass m but not with its own weight, to make sense of the function f. – Mike Shulman Mar 30 '14 at 14:32
@MikeShulman: I am not certain whether I understand your problem. The usual way to burden a rope is by attaching a piece of weight to it. Also, using forces may be clearer, but adds another physical concept, which is why I did not do that. Finally, I rephrased, what I thought to be the phrase in question. – Wrzlprmft Mar 30 '14 at 18:37
The rephrasing helps a bit. Maybe my problem is that the very phrase "If a total of mass m is pulling on a piece of rope, it extends to f(m) of its unburdened length" is only literally true when the "piece of rope" is infinitesimal. Right? – Mike Shulman Mar 30 '14 at 20:37
@MikeShulman: Yes. Unless the rope is not oriented vertically, but horizontally (and the force of the mass pulling is mediated by a weigth-less rope and redirected via a wheel, e.g.). – Wrzlprmft Mar 30 '14 at 20:42

Chemical reaction rates and how they relate to changes in temperature.

I use this example when teaching chemical engineering students about integrals. The Boltzmann distribution describes the velocity distribution of molecules in a gas.

$$f(v) = \sqrt{\frac{2}{\pi} \left( \frac{m}{k_B \cdot T} \right)^3} \cdot v^2 \cdot \exp\left( \frac{-m v^2}{2 k_B T} \right)$$

The integral $\int_a^b f(v) dv$ is then the fraction of the gas molecules with a velocity in the range $[a, b]$. For a certain reaction to occur there must be a minimal amount of kinetic energy present (the activation energy, $E_a$), from that the lowest possible velocity of a gas molecule with this kinetic energy can be determined by

$$v_\textrm{min} = \sqrt{\frac{2 E_a}{m}}$$

The fraction of gas molecules that can participate in the reaction is then given by

$$\int_{v_\textrm{min}}^{\infty} f(v) \, dv$$

This can be used to investigate how changes in temperature alters the fraction of gas molecules that have sufficient kinetic energy to overcome the activation energy.

One application is to test the rule of thumb, which states that an increase in temperature of 10 kelvin will roughly double the reaction rate.

Below is Mathematica code for solving this

assump = Assumptions -> {k > 0, T > 0, 
   m > 0}; MaxwellBoltzmannSpeedDistribution =
 Sqrt[2/Pi (m/(k T))^3] v^2 Exp[-m v^2/(2 k T)];
FractionOfParticlesWithEnergyAbove = 
   MaxwellBoltzmannSpeedDistribution, {v, minVelocity, Infinity}, 
TemperatureDependency = 
 FractionOfParticlesWithEnergyAbove //. {minVelocity -> 
    Sqrt[2 10^-19/m], k -> 1.3806488*10^(-23), m -> 1.6726 10^-27}
res = Table[{T, TemperatureDependency} /. {T -> Tval}, {Tval, 293, 
    343, 10}];
res // TableForm

The above code gives the following output

293   1.05132*10^-10
303   2.33909*10^-10
313   4.94258*10^-10
323   9.96656*10^-10
333   1.926*10^-9
343   3.58024*10^-9
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Amazing! As a math educator who has never seen this equation before, what would be some good values of those constants m and k_B for a real-world example? – Chris Cunningham Apr 2 '14 at 15:05
The missing values are: $m = 1.67\cdot 10^{-27} kg$ (atom mass of hydrogen) and $k_B = 1.38\cdot 10^{-23} \cdot m^2 \cdot kg \cdot s^{-2} \cdot K^{-1}$ (Boltzmanns constant). – midtiby Apr 3 '14 at 6:04

Let them guess the mass of a mountain by showing them some pictures of it. They then can then model the mountain in different ways:

  • cone
  • rotational body
  • irregular pyramid

Additionally they can model the tip of the mountain in different ways

  • flat
  • flat with inverse semi-ball for the caldera
  • flat with inverse cone for the caldera

You can also give them elevation data and let them model the mountain as a stack of layers.

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Newton's law of cooling: The rate at which the cup of tea cools is proportional to it's temperature difference to the environment.

Compute the areas/volumes of curved figures/solids. I.e., the area under a parabola, the area of a circle, the volume of a cone, a cylinder, a sphere. Can even introduce/prove Cavalieri's principle.

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Doesn’t Newton’s law of cooling (or any cooling) lead to a differential equation? – Wrzlprmft Mar 29 '14 at 23:36
@Wrzlprmft, yes, but one easily solved (it is just $y'(t) = -a(y - y_\infty)$, with $y(0) = y_0$) – vonbrand Mar 29 '14 at 23:59
Your examples of areas and volumes are all easily computed with the standard textbook formulas. – Mike Shulman Mar 30 '14 at 14:17
@MikeShulman, I understood the question was on ways to show how integration is used, in a setting different from "compute $\int_a^b f(x) \, \mathrm{d} x$, where $a$, $b$, $f$ are given." To get the point of what it is used for across, it is best if the computation of the resulting integral is simple. You don't want a series of complex maneuvers to obscure that. – vonbrand Mar 30 '14 at 23:53
It has nothing to do with simplicity or complexity of the resulting integral. I don't care whether the resulting integral is even doable by elementary means or not; mainly I want them to be able to set up integrals correctly in situations that don't fit neatly into formaulas that they've memorized. – Mike Shulman Mar 31 '14 at 21:00

This example is actually from a textbook I am using.

You are in a submarine and want to know how long you have gone in some time (there are no km marks to read under water!). You have a speedometer, so you can just write down the speedometer writings, and integrate them numerically with respect to time.

(You can do that in a car too, but ther there is no øpoint in doing it).

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This sort of example is good, but for me it belongs to a different part of calculus, when we are learning the meaning of integrals and how they relate to Riemann sums. – Mike Shulman Mar 31 '14 at 21:02

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