Editing Functional (mathematics) (section)

===Definite integral===
[[Integral]]s such as
<math display="block">f\mapsto I[f] = \int_{\Omega} H(f(x),f'(x),\ldots) \; \mu(\mathrm{d}x)</math>
form a special class of functionals. They map a function <math>f</math> into a real number, provided that <math>H</math> is real-valued. Examples include
* the area underneath the graph of a positive function <math>f</math> <math display=block>f\mapsto\int_{x_0}^{x_1}f(x)\;\mathrm{d}x</math>
* [[Lp norm|<math>L^p</math> norm]] of a function on a set <math>E</math> <math display=block>f\mapsto \left(\int_E|f|^p \; \mathrm{d}x\right)^{1/p}</math>
* the [[arclength]] of a curve in 2-dimensional Euclidean space <math display=block>f \mapsto \int_{x_0}^{x_1} \sqrt{ 1+|f'(x)|^2 } \; \mathrm{d}x</math>