Functions of the form $\int_a^x f(t) dt$ that are commonly used.

Question

I am a graduate student and teaching assistant, and I am teaching Calc 1 for the first time. In a few weeks I will be covering the Fundamental Theorem of Calculus. I'm using James Stewart's Calculus textbook, and I was hoping to give students several "real world" examples of functions of the form $g(x)=\int_a^x f(t) dt$ to make the first part of the FTC more accessible. Stewart gives one example, the Fresnel function

$$S(x)=\int_0^x \sin (\pi t^2/2)$$

which is (apparently) used in the theory of the diffraction of light waves. But I was hoping for more examples from physics, chemistry, etc. Any thoughts or ideas?

Answer 1

$$ \Phi(z) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^x e^{-z^2/2}\,dz $$

This is the cumulative distribution function of the standard normal distribution, seen in every course on statistics. Physicists often talk about the "error function", in which $z^2$ appears instead of $z^2/2$ (and then the normalizing constant is different). That's trivially equivalent to this one, but with this version, the standard deviation is $1$.

Abraham de Moivre considered this function in the 18th century while thinking about a coin-tossing problem. Suppose you toss a coin 1800 times. What is the probability that the number of heads is in the range (say) $\{886,\ldots, 908\}$? An exact answer is computationally far too expensive, but the integral above does it.

The derivative of $\Phi$ is conventionally denoted $\varphi$.

The one seen tabulated in the back of every statistics textbook is $\Phi$.

Answer 2

In cartography in the year 1599 the following problem arose. Mercator wanted a map of the world on which compass bearings on the earth correspond to those on the map. E.g. $13^\circ$ east of north on the surface of the earth corresponds to $13^\circ$ counterclockwise from straight up on the map, at every point. Going through a bit of geometry, it is found that at latitude $\theta$, the ratio of distance on the map to distance on the earth has to be proportional to $\sec\theta$. That implies that the location of points on the map at latitude $\theta$ should be at the following distance from the equator: $$ \int_0^\theta \sec\alpha\,d\alpha. $$ Evaluating this integral remained a famous unsolved problem for (I think?) more than half a century.

Answer 3

The incomplete Beta function $$ x\mapsto \frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)} \int_0^x u^{\alpha-1}(1-u)^{\beta-1}\,du\quad\text{ for }0\le x\le 1 $$ arises in probability and statistics. When the two parameters $\alpha$, $\beta$ are positive integers, then the incomplete Beta function is a polynomial function, but it probably cannot be expressed more simply than by this integral.

Answer 4

The error function $\dfrac{2}{\sqrt{\pi}} \int_0^x e^{-t^2}\ dt$ has many applications, especially in probability and statistics, and from there to physics.

Answer 5

In quantum mechanics the method of time dependent perturbation theory yield integrals like this. For instance the probability amplitude for a transition from the ground state of an atom to some higher excited state due to an eternally applied oscillating electric field is given by an integral like,

$$ \phi_{0 \rightarrow n} = \frac{1}{i\hbar} \int_0^T H'_{0n} e^{i\omega_{0n} t} dt $$

Answer 6

In probability, if the random variable is continuous, rather than discrete, then

$P(a \leq X \leq b) = \int_{a}^{b} f(x)\,dx$ where $f$ has a lebesgue integral.

The cumulative distribution function to a continuous random variable is $F(x)=\int_{-\infty}^x f(u)\,du$

This is very widely used in just about any question or real life scenario where continuous probability is involved.

The very best of luck on your new job.

Answer 7

Sine integral has such form:

$$\operatorname{Si}(x)=\int_0^x\frac{\sin t}{t}\text{d}t.$$

Similar form is for the cosine integral, although it's not quite the same. See also logarithmic integral, exponential integral, I'm sure you could find more.

Answer 8

Why do you need any examples?

In my opinion, FTC is the most important part in Calculus because how it unites the other two parts (dir/int).

The derivative-form of FTC implies that the overall properties of a function (integral) is decided by the domestic properties (derivative).

The integral-form of FTC of FTC indicates that the domestic properties are regulated by the overall properties.

With FTC, it's much easier to do integral with anti-derivatives. It's an example to tell the students how the advancements in mathematics make it easier to solve problems.

I don't think it's needed to make the "first part" of FTC accessible. Actually FTC is FTC and the two so-called parts are essentially the same. They are just represented in two different forms. Similarly, differentiation and integration are essentially the same. So do the two mean value theorems.