Functions of the form $\int_a^x f(t) dt$ that are commonly used. 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?
 A: $$
\Phi(z) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^x e^{-z^2/2}\,dz
$$
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 $\Phi$.
A: 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.
A: 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.
A: 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.
A: 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,
$$ \phi_{0 \rightarrow n} = \frac{1}{i\hbar} \int_0^T H'_{0n} e^{i\omega_{0n} t} dt $$
A: 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.
A: 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.
A: 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.
