# Integral **The continuous analogue of a sum.** ![[Integral.svg|500]] *A definite integral can be represented as the total signed area of the region between the graph of a function and the $x$-axis on the closed interval* An **integral** is the continuous equivalent to a sum of discrete values. It may also refer to an [[antiderivative]], in which case they are also known as **indefinite integrals**. A **definite integral** is the *signed area* of the region bounded by the graph of a [[function]] and the $x$-axis between two points on the real line. The process of determining an integral is *integration* and is the inverse of [[Derivative|differentiation]]. Integration and differentiation are related by the [[fundamental theorem of calculus]]. ## Definition ### Riemann integral > [!Riemann integral] > Let $f$ be a *bounded* real-valued function that is *continuous almost everywhere* on a *closed interval*. > > $f$ is *Riemann-integrable* on the interval if the limit of the [[Riemann integral#Riemann sum|Riemann sums]] > $\sum_{i=0}^{n-1}f(t_{i})\Delta_{i}$ > of $f$ exists as the size of the sub-intervals $\Delta_{i}$ approaches zero. > > This limit is the [[Riemann integral]] of $f$ on the interval. ![[RiemannIntegral.svg]] *A sequence of Riemann sums converging towards the integral as the width of the partitions approach zero; the limit of the sequence is the Riemann integral* A proper Riemann integral is neither able to integrate *unbounded* functions nor functions with an *uncountable* number of [[Discontinuity|discontinuities]]. Unbounded functions require [[Improper Integral|improper integrals]], an extension of definite integrals, to compute. Functions with many discontinuities are computable using *Lebesgue integrals*. ### Lebesgue integral > [!Lebesgue integral] > Let $f$ be a *non-negative* real-valued function defined on a *closed interval*. > > $f$ is *Lebesgue-integrable* on the interval if the limit of > $\sum_{i=0}^{n-1}\Delta_{i}\;\mu(\{x\mid f(x)>y_{i}\})$ > exists as the size of the [[Image|range]] partitions $\Delta_{i}$ approaches zero, where $\mu$ is the [[Lebesgue Integral#Lebesgue measure|Lebesgue measure]]. > > This limit is the [[Lebesgue integral]] of $f$ on the interval. ![[LebesgueIntegral.svg]] *A sequence of simple function approximations converging towards the integral as the size of the range partitions approach zero; the limit of the sequence is the Lebesgue integral*[^1] #### Signed functions The Lebesgue integral of some *signed* real-valued function $f$ can be found by splitting the function into two non-negative functions, $f^{+}$ and $f^{-}$. $f^{+}(x)=\begin{cases}f(x)\quad &\text{if } f(x)> 0, \\ 0 \quad&\text{otherwise}\end{cases}$ $f^{-}(x)=\begin{cases}-f(x)\quad &\text{if } f(x)< 0, \\ 0 \quad&\text{otherwise}\end{cases}$ The function can then be represented as $f=f^{+}-f^{-}$ > [!NOTE] Lebesgue integral of a signed function > If it exists, the Lebesgue integral of a *signed function* $f$ is > $\int f=\int f^{+}-\int f^{-}$ ## Notation The notation for a definite integral of the function $f(x)$ with respect to $x$ over the interval $[a,b]$ is $\int_{a}^{b}f(x)\;dx$ The function is known as the *integrand* and $a$ and $b$ are known as the *limits* or *bounds of integration*. The interval $[a,b]$ is also known as the *interval of integration*. If the integral is indefinite then the integral is notated without limits. $\int f(x)\;dx$ ## Properties ### Linearity of integration The integral of a *linear combination* of functions is equal to the same linear combination of their individual integrals. > [!NOTE] Multiplication by a scalar > $\int_{a}^{b}\alpha f(x)\;dx=\alpha\int_{a}^{b}f(x)\;dx$ > [!NOTE] Addition > $\int_{a}^{b}f(x)+g(x)\;dx=\int_{a}^{b}f(x)\;dx+\int_{a}^{b}g(x)\;dx$ ### Inequalities > [!NOTE] Inequalities between functions > If $f(x)\le g(x)$ for all $x$ on the interval $[a,b]$, then > $\int_{a}^{b}f(x)\;dx\le\int_{a}^{b}g(x)\;dx$ ### Degenerate intervals > [!NOTE] Reversed interval > If $f(x)$ is a real-valued function defined over a closed interval $[a,b]$ such that $a < b$, then > $\int_{b}^{a}f(x)\;dx=-\int_{a}^{b}f(x)\;dx$ > [!NOTE] Integral over a point > If $f(x)$ is a real-valued function defined on a point $a$, then > $\int_{a}^{a}f(x)\;dx=0$ ## Extensions ### Improper integral An [[improper integral]] is an extension of Riemann integrals in which either the integrand function or limits of integration are *unbounded* or the integrand contains a *discontinuity*. An improper integral is actually the limit or the sum of limits of a definite integral. Thus, improper integrals are described as either convergent or divergent. An improper integral has a value if and only if it converges, although a [[Cauchy principal value]] can be assigned to divergent integrals. Improper integrals over unbounded limits are of any of the following forms: $\int_{a}^{\infty}f(x)\;dx\quad\int_{-\infty}^{b}f(x)\;dx\quad\int_{-\infty}^{\infty}f(x)\;dx$ ![[ImproperIntegralUnboundedLimit.svg|550]] *Improper integral with unbounded limits of integration* ![[ImproperIntegralUnboundedFunction.svg|400]] *Improper integral of an unbounded function* ### Multiple integral A [[multiple integral]] is the generalisation of a definite integral to functions of *multiple variables*. In a multiple integral, the interval of integration is replaced with the *domain of integration*. A double integral, for example, is an integral equal to the *signed volume* between the surface of a multivariable function and the plane on which the domain of integration is defined. $\int_{D} f(x,y)\;dA=\iint_D f(x,y)\;dA=\int_{a}^{b}\left[\int_{c}^{d}f(x,y)\;dy\right]\;dx$ ![[DoubleIntegral.svg|500]] *A double integral* ### Line integral A [[line integral]] is a generalisation of the integration of multivariable functions, typically scalar or vector fields, onto *curves* as the domain of integration. The following are the notations for a general line integral over a scalar and a vector field, respectively: $\int_C f\;ds=\int _{a}^{b}f(\mathbf{r}(t))|\mathbf{r}'(t)|\;dt$ $\int_C \mathbf{F}(\mathbf{r})\cdot d\mathbf{r}=\int_{a}^{b}\mathbf{F}(\mathbf{r}(t))\cdot\mathbf{r}'(t)\;dt$ ![[LineIntegral.svg|500]] *A line integral of a scalar field* #### Contour integral A [[Contour Integral|contour integral]] is a generalisation of the line integral to integrating *complex functions* along contours in the [[Complex Number\|complex plane]]. Often, the contour on which the integral is taken is *closed* and *orientated* which may be notated with if a circle intersecting the integral sign. $\int_{\gamma}f(z)\;dz\quad\oint_{\gamma}f(z)\;dz$ ![[ContourIntegral1.svg|600]] ![[ContourIntegral2.svg|500]] *A contour integral; the contour of integration (above) and the modulus of the integrand function (below) are shown* ### Surface integral A [[surface integral]] is a generalisation of multiple integrals onto *surfaces* as the domain of integration. It is analogous to a line integral where the domain of integration is *two-dimensional* instead. The following are the notations for a general surface integral over a scalar and a vector field, respectively: $\iint_{S}f\;dS=\iint_{T}f(\mathbf{r}(u,v))\left\|{\frac{\partial\mathbf{r}}{\partial u}\times\frac{\partial\mathbf{r}}{\partial v}}\right\|\;du\;dv$ $\iint_{S}\mathbf{F}\cdot d\mathbf{s}=\iint_{T}\mathbf{F}(\mathbf{r}(u,v))\cdot\left(\frac{\partial\mathbf{r}}{\partial u}\times\frac{\partial\mathbf{r}}{\partial v}\right)\;du\;dv$ ![[SurfaceIntegral.svg|600]] *A surface of integration divided into differential patches* [^1]: The Lebesgue integral is only defined for *non-negative* functions, so only the positive part of the function is integrated here.