Definite Integrals
The red area is above the axis and is positive. The blue area is below the axis and is negative.
A definite integral is a formal calculation of area beneath a function, using infinitesimal slivers or stripes of the region. Integrals may represent the (signed) area of a region, the accumulated value of a function changing over time, or the quantity of an item given its density. They were first studied by \(17^\text{th}\)-century mathematicians Isaac Newton and Gottfried Liebniz, who independently developed their own systems of integration. The modern notation follows from Liebniz's notes, and given a real-valued function \(f\) and real numbers \(a < b\), the definite integral is written
\[ \int_a^b f(x) \, dx.\]
This value represents the signed area between the function \(f\), the \(x\)-axis, and the lines \(x = a\) and \(x = b;\) regions above the \(x\)-axis have positive area, while regions below the \(x\)-axis have negative area.
Definite integrals have an indefinite form as well that serves as a partial inverse to differentiation. Just as differentiation measures a function's incremental changes, a definite integral attempts to "un-do" that. So integrals focus on aggregation rather than change.
Definite integrals are useful in economics, finance, physics, and engineering. For instance, marginal cost accrues to cost, income rates accrue to total income, velocity accrues to distance, and density accrues to volume. Definite integrals are also used to perform operations on functions: calculating arc length, volumes, surface areas, and more. Line integrals, surface integrals, and contour integrals are examples of definite integrals in generalized settings.
Definition
The initial definition, provided by Bernhard Riemann, expresses area as a combination of infinitely many vertically-oriented rectangles, a technique known as Riemann sums. One benefit of this definition is that it is visually intuitive.
As the rectangles get thinner, the combined area of the rectangles approaches the value of the integral.
An integral is composed of three items of information: the limits, the integrand, and the differential:
\[\int_a^b f(x) \, dx.\]
The limits are \(a\) and \(b\), the integrand is \(f(x)\), and the differential is \(dx\). The limits of integration give information about where the integration takes place, and the interval of integration is the interval \([a, \, b]\) determined by these limits. The integrand gives information about the shape of the region and represents the height of each rectangle in the Riemann sum. The differential gives information about what variable the integrand uses and represents the width of each rectangle in the Riemann sum.
By the fundamental theorem of calculus, given a function \(f\) defined on the interval \([a, \, b]\) with antiderivative \(F\),
\[ \int_a^b f(x) \, dx = F(b) - F(a).\]
This translates the language of integration from something purely geometric into a structured algebraic construct that may be manipulated in various ways.
Properties
Being one of the fundamental tools of calculus, integrals have a large number of properties coming from the geometry of the coordinate plane, the definition of a functional, and the relationship between integrals and derivatives. Integrals also have an algebraic interpretation that allows for very useful techniques like \(u\)-substitution that are necessary for many types of integral evaluation (and in proofs of many properties below!).
Sometimes, it is possible to relate the area of one integral to that of others.
Some properties give information about the limits.
The interval \([a, \, b]\) is by convention written as \(\int_a^b\). When reversing the order of \(a\) and \(b\), the integral interprets the limits as a "negative" interval:
\[\displaystyle \int _{ a }^{ b }{ f(x)\, dx } =- \displaystyle \int _{ b }^{ a }{ f(x)\, dx }. \]
Just as an interval along the real line may be split up, intervals of integration may be split up in accordance with the geometric intuition. For any value \(c \in [a, \, b]\),
\[\displaystyle \int _{ a }^{ b }{ f(x)\, dx } =\displaystyle \int _{ a }^{ c }{ f(x)\, dx } + \displaystyle \int_{ c }^{ b }{ f(x)\, dx }. \]
In fact, the previous property allows for \(c\) to take any value without the risk of ambiguity.
When the function \(f\) is defined piecewise in \([a,b]\), the integral may be evaluated by breaking up the interval of integration into several sub-intervals so that the function is continuous and easily defined in each sub-interval.
Evaluate \(\displaystyle \int _{ -4 }^{ 3 }{ \big|x^2-4\big|\, dx }.\)
We have
\[\begin{align} \int _{ -4 }^{ 3 }{ \big|x^2-4\big|\, dx } &=\int _{ -4 }^{ -2 }{ \big|x^2-4\big|dx }+\int _{ -2 }^{ 2 }{ \big|x^2-4\big|dx }+\int _{ 2 }^{ 3 }{ \big|x^2-4\big|dx }\\ &=\int _{ -4 }^{ -2 }{ \big(x^2-4\big)dx }+ \int _{ -2 }^{ 2 }{ \big(4-x^2\big)dx }+\int _{ 2 }^{ 3 }{ \big(x^2-4\big)dx }\\ &=\left[\frac{x^3}{3}-4x\right]_{ -4 }^{ -2 }+\left[4x-\frac{x^3}{3}\right]_{ -2 }^{ 2 } +\left[\frac{x^3}{3}-4x\right]_{ 2 }^{ 3 }\\ &=\frac{71}{3}.\ _\square \end{align}\]
Some properties give information about the integrand.
As an operator, the integral is linear. That is, for any real numbers \(c\) and \(d\) and functions \(f\) and \(g\),
\[ \int_a^b c \cdot f(x) + d \cdot g(x) \, dx = c \int_a^b f(x) \, dx + d \int_a^b g(x) \, dx.\]
An integral may be computed with Riemann sums summed from the left just as easily as it may be computed with Riemann sums summed from the right. Algebraically, this is equivalent to saying
\[\displaystyle \int _{ a }^{ b }{ f(x)\, dx } =\displaystyle \int _{ a }^{ b }{ f(a+b-x)\, dx }. \]
Let \(\displaystyle I=\int _{ a }^{ b }{ f(x)\, dx }.\) Also, let \(x=a+b-t \implies dx=-dt.\) Then
\[\begin{align} I &= \int _{ b }^{ a }{ f(a+b-t)\, (-dt) }\\ &= \int _{ a }^{ b }{ f(a+b-t)\, dt } &\qquad (\text{by Property 2})\\ &=\int _{ a }^{ b }{ f(a+b-x)\, dx}.\ _\square \end{align}\]
Evaluate \(\displaystyle \int _{ \pi /6 }^{ \pi/3 }{ \frac{dx}{1+\tan^5x} }. \)
Using
\[\displaystyle \int _{ a }^{ b }{ f(x)dx } =\displaystyle \int _{ a }^{ b }{ f(a+b-x)dx }, \]
we have
\[\begin{align} I &= \int _{ \pi /6 }^{ \pi/3 }{ \frac{dx}{1+\tan^5x} } \\ &= \int _{ \pi /6 }^{ \pi/3 }{ \frac{dx}{1+\tan^5\left(\frac{\pi}{3}+\frac{\pi}{6}-x\right)} }\\ &= \int _{ \pi /6 }^{ \pi/3 }{ \frac{dx}{1+\cot^5x} }\\ \\ \Rightarrow 2I &= \int _{ \pi /6 }^{ \pi/3 }{ \frac{dx}{1+\cot^5x} }+\int _{ \pi /6 }^{ \pi/3 }{ \frac{dx}{1+\tan^5x} }\\ &=\int _{ \pi /6 }^{ \pi/3 } dx\\ &=x\Big| _{ \pi /6 }^{ \pi/3 }\\ \\ \Rightarrow I&=\frac{\pi}{12}. \ _\square \end{align}\]
Similarly, an integral may be computed by calculating area from the left and from the right simultaneously (and stopping in the middle):
\[\displaystyle \int _{ a }^{ b }{ f(x)\, dx } =\displaystyle \int _{ a }^{ (a+b)/2 }{\big( f(x)+f(a+b-x)\big)\, dx. } \\\]
For an integer \(n\), evaluate \(\displaystyle \int _0^{\pi}e^{\cos^2x}\cos^3\big((2n+1)x\big)\, dx.\)
We have
\[\begin{align} I&=\int_0^{\pi} e^{\cos^2x}\cos^3\big((2n+1)x\big)\, dx\\ &=\int_0^{\pi/2}\left[ e^{\cos^2x}\cos^3\big((2n+1)x\big)+ e^{\cos^2(\pi-x)}\cos^3\big((2n+1)(\pi-x)\big)\right]dx\\ &=\int_0^{\pi/2} e^{\cos^2x}\left[\cos^3\big((2n+1)x\big)-\cos^3\big((2n+1)x\big)\right]dx\\ &=0.\ _\square \end{align}\]
It follows that, for all odd functions \(f,\) we have \( \displaystyle{\int_{-a}^a f(x) \, dx = 0} \).
Other properties give information about how to change the integrand. Changing the variable of integration has no effect upon the value of the integral:
\[\displaystyle \int _{ a }^{ b }{ f(x)\, dx } =\displaystyle \int _{ a }^{ b }{ f(z)\, dz }. \]
In fact, a \(u\)-substitution \(u = g(x)\) may transform the integral, provided \(g\) is differentiable, \(h = g^{-1}\) is defined in the interval \([g(a), \, g(b)]\), and \(\displaystyle{\lim_{t \to u} \tfrac{f\big(h(t)\big)}{g'(t)}}\) exists for all \(u \in [g(a), \, g(b)]:\)
\[ \int_a^b f(x) \, dx = \int_{g(a)}^{g(b)} \frac{f\big(h(u)\big)}{g'(u)} \, du. \]
Note that, for instance, one of the properties in the previous subsection follows from the \(u\)-substitution \(u = a + b - x\).
\[\] Finally, some properties compare the values of different integrals. Many of these extend the intuition for thinking of an integral as a type of summation.
If \(f(x) < g(x)\) for all \(x\) in \([a, \, b]\), then
\[ \int_a^b f(x) \, dx < \int_a^b g(x) \, dx. \]
This property is consistent with the idea of an integral as a measure of area.
Both the Cauchy-Schwarz inequality and Holder's inequality hold for integrals as well as summations. For instance, the Cauchy-Schwarz inequality for integrals states that
\[ \left( \int_a^b f(x) g(x) \, dx \right)^2 \le \left( \int_a^b f(x)^2 \, dx \right) \cdot \left( \int_a^b g(x)^2 \, dx \right). \]
The Minkowski inequality, which is crucial in the development of real analysis and integration theory, states that for any real number \(p > 1\)
\[ \left( \int_a^b \big|f(x) + g(x)\big|^p \, dx \right)^{1/p} \le \left( \int_a^b \big|f(x)\big|^p \, dx \right)^{1/p} + \left( \int_a^b \big|g(x)\big|^p \, dx \right)^{1/p}. \]
Different Types of Integrals
Just as a definite integral can be over an interval with finite limits, it can also be defined over any interval of the real line — including those with infinite limits. For any real number \(a\) and function \(f\),
\[ \int_a^\infty f(x) \, dx := \lim_{b \to \infty} \int_a^b f(x) \, dx. \]
Intervals of the form \((-\infty, b]\) can be similarly defined. These are called improper integrals.
In addition to single-variable integrals, there are line integrals and contour integrals for functions of multiple variables. A line integral is a function \(g(x, \, y)\) that can be integrated over a curve \(\gamma(t) = \big(x(t), \, y(t)\big)\) from \(t=a\) to \(t=b\). It is equal to
\[ \int_\gamma g \, ds = \int_a^b g\big(x(t), \, y(t)\big) \sqrt{\big(x'(t)\big)^2 + \big(y'(t)\big)^2} \, dt.\]
A surface integral is similar to a line integral — but for double integrals. Integrals may be nested within each other in order to integrate over multiple dimensions (like with a surface).
There are also methods of integration over complex numbers and more exotic number fields. Measure theory extends the concept of an interval of integration to any set satisfying a collection of parameters. It also provides a way of defining different methods of integration than the one proposed by Riemann that allow for integration over more generalized sets (e.g., vector spaces of functions).
Methods of Integration
A great deal of integration tricks exist for evaluating definite integrals exactly, but there still exist many integrals for each of which there does not exist a closed-form expression in terms of elementary mathematical functions. For instance, the integral
\[ \int_0^1 e^{x^2} \, dx\]
may not be evaluated without numerical methods.
Many numerical methods exist to approximate integrals to any degree of accuracy, but their efficiency is wholly dependent upon the function and interval in question. When possible, an exact calculation is always preferred.
Partial Riemann sums, the trapezoid rule, and Simpson's rule all seek to provide a geometric approximation of the region in question. Other approaches, like those using Chebyshev's formula, seek to model the function in question (at least, in the relevant interval) with functions that are easier to integrate.
Determining good approximations for definite integrals is one of the principal aims of numerical analysis.