Literature
Functionals [Video lecture]
A functional assigns a number to each function in a class. Thus, it is a function where the independent variable is itself a function.
Examples
$$ J[y] = \int_{x_1}^{x_2} y(x)\, dx $$
gives the area between the $x$ axis and the graph of the function.
$$ J[y] = \int_a^b y'^2(x)\,dx. $$
$$ J[y] = \int_a^b F(x,y,y')\,dx.\; \square $$