$\renewcommand{\Re}{\operatorname{Re}}$ $\renewcommand{\Im}{\operatorname{Im}}$ $\newcommand{\erf}{\operatorname{erf}}$ $\newcommand{\dag}{\dagger}$ $\newcommand{\const}{\mathrm{const}}$ $\newcommand{\arcsinh}{\operatorname{arcsinh}}$ $\newcommand{\bfnu}{\boldsymbol{\nu}}$
There are several classical problems.
Problem 1. We need to construct the fastest slide from point $(0,0)$ to $(a,-h)$. If $y(x)$ describes its shape then time is \begin{equation} T= \int_0^a \frac{1}{\sqrt{2guy}} \sqrt{1+y^{\prime\,2}}\,dx. \label{eq-10.2.P.1} \end{equation} We need to minimize $T$.
Problem 2. The heavy flexible but unstretchable wire (chain) has a length and an energy respectively \begin{gather} L= \int_0^a \sqrt{1+y^{\prime\,2}}\,dx, \label{eq-10.2.P.2}\\ U=\rho g \int_0^a y \sqrt{1+y^{\prime\,2}}\,dx \label{eq-10.2.P.3} \end{gather} where $\rho$ is a linear density. We need to minimize energy energy $U$ as length $L$ is fixed.
Problem 3.
Consider curve $y=y(x)$. The length of it and the area bounded by it are respectively
\begin{gather}
L=\int_0^a \sqrt{1+y^{\prime\,2}}\,dx,\\
A=\int_0^a ydx.
\end{gather}
We need to maximize $A$ as length $L$ is fixed.
Write Euler-Lagrange equation and integrate it.
Problem 4. Consider curve $y=y(x)$. The area of the surface of the revolution formed by it and the volume inside of it are respectively \begin{gather} S=2\pi\int_0^a y\sqrt{1+y^{\prime\,2}}\,dx,\\ V=\pi\int_0^a y^2\,dx. \end{gather}
Problem 5. If in 2D–light propagation the speed of light at point $(x,y)$ is $c(x,y)$ then the time of travel between two points $(x_1,y_1)$ and $(x_2,y_2)$ equals \begin{equation} T=\int_{x_1}^{x_2} \frac{\sqrt{1+y'^2}\,dx}{c(x,y(x))} \end{equation} where $y=y(x)$ is equation of the ray, $u(x_j)=y_j$, $j=1,2$.
Problem 6. We need to minimize \begin{equation} \int_0^a (u''^2 -2 u)\,dx \end{equation} under constrains below.
Problem 7. We need to minimize \begin{equation} \int_0^a u''^2 \,dx \end{equation} under constrain \begin{equation} \int_0^a u^2 \,dx=1 \end{equation} and constrains below.