# Fundamental Mathematics/Arithmetic/Integration

Integration is a mathematic operation on a function to find area under the function's curve.

Integration is denoted as

${\displaystyle \int f(t)dt}$

An integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data. Integration is one of the two main operations of calculus, with its inverse, differentiation, being the other.

Given a function f of a real variable x and an interval [a, b] of the real line, the definite integral is defined informally as the signed area of the region in the xy-plane that is bounded by the graph of f, the x-axis and the vertical lines x = a and x = b. The area above the x-axis adds to the total and that below the x-axis subtracts from the total.

Roughly speaking, the operation of integration is the reverse of differentiation. For this reason, the term integral may also refer to the related notion of the antiderivative, a function F whose derivative is the given function f. In this case, it is called an indefinite integral.

## Types of integration

### Definite integral

${\displaystyle \int f(x)\,dx}$
The integral sign represents integration
dx, called the differential of the variable x, indicates that the variable of integration is x.
f(x) the function to be integrated is called the integrand.
dx is separated from the integrand by a space (as shown).

If a function has an integral, it is said to be integrable. The points a and b are called the limits of the integral. An integral where the limits are specified is called a indefinite integral. The integral is said to be over the interval [a, b]

${\displaystyle \int f(x)\,dx=F(x)+C}$

### Indefinite integral

When the limits are omitted, as in the integral is called an indefinite integral, which represents a class of functions (the antiderivative) whose derivative is the integrand.

${\displaystyle \displaystyle \int _{a}^{b}f(x)\,dx}$.

The fundamental theorem of calculus relates the evaluation of definite integrals to indefinite integrals. Occasionally, limits of integration are omitted for definite integrals when the same limits occur repeatedly in a particular context. Usually, the author will make this convention clear at the beginning of the relevant text.

There are several extensions of the notation for integrals to encompass integration on unbounded domains and/or in multiple dimensions (see later sections of this article). The integral with respect to x of a real-valued function f(x) of a real variable x on the interval [a, b] is written as

${\displaystyle \int _{a}^{b}\!f(x)\,dx=F(b)-F(a).}$

## Integral transformation

### Laplace transformation

Transform function in time domain into complex domain

${\displaystyle F(s)=L[f(t)]=\int _{0}^{\infty }f(t)e^{-st}\,dt}$

Where s is a complex number frequency parameter ${\displaystyle s=\sigma +i\omega }$, with real numbers σ and ω.

Example

${\displaystyle F(s)=L[{\frac {d}{dt}}]=s}$
${\displaystyle F(s)=L[\int dt]={\frac {1}{s}}}$

### Fourier transformation

Transform function in time domain into frequency domain

${\displaystyle F(j\omega )=\int _{0}^{\infty }f(t)e^{-j\omega t}\,dt}$

Example

${\displaystyle F(j\omega )=F[{\frac {d}{dt}}]=j\omega }$
${\displaystyle F(j\omega )=F[\int dt]={\frac {1}{j\omega }}}$

### Applications

Voltage of capacitor in time domain

${\displaystyle v={\frac {1}{C}}\int idt}$

Transform into complex domain

${\displaystyle v={\frac {1}{sC}}}$

Transform into frequency

${\displaystyle v={\frac {1}{j\omega C}}}$