Integral

The integral is a central concept in calculus. Intuitively, we can think of an integral as a measure of the totality of an object with an extent in space. For example, integral calculus lets us calculate the length of a curve, the area of a surface, or the volume of a solid object. In other contexts, an integral might measure one quantity that depends on another important quantity that is varying: the distance that a rocket has traveled, for example, depends upon its acceleration which is varying as the rocket's mass decreases from fuel usage, and an integral can take this complication into account. Finally, within calculus integration (the process of calculating integrals) can be seen as an inverse operation to differentiation, and so integrals are of great use in the many contexts where derivatives are involved.

A geometric definition

The easiest way to understand integrals is perhaps as a means to calculate area. What do we mean by area in the first place? We do know the precise meaning of area in the case of one simple figure: the rectangle. A rectangle that is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w} units wide and $h$ units high has area Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w \times h} ; let us take this as the definition of area, along with the property that the cumulative area of two rectangles next to each other is the sum of their respective areas. We can now measure the area of a more complicated shape, such as an apartment floor, by covering it with rectangles, and taking the sum of their individual areas. This is the basic meaning of integration: an integral is simply a sum of smaller parts that together add up to the whole.

Walls are typically at right angles, so tiling a floor with rectangles is no problem. But there are infinitely many kinds of shapes that cannot be exactly covered with rectangles, such as circles, ellipses, or the interior of any curved shape we can draw. Nevertheless, we think of these shapes as having area. We can approximately measure the area of such a shape by covering it with many small rectangles. The more and smaller rectangles we choose, the better the approximation becomes. Using the concept of a limit from mathematical analysis, we can continue to shrink the rectangles until they become infinitely small and the error becomes zero. This process of taking limits is what distinguishes integrals from ordinary sums, and it allows us to exactly calculate lengths, areas, volumes — and so on, of arbitrarily complicated shapes, provided of course that we can express those shapes with exact mathematical formulas.

Let us now give a more formal definition of integral, and also introduce the mathematical notation. Consider a region in the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} -$y$-plane delimited by the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} -axis, two vertical lines at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=a} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=b} , and a curve described by the function $y=f(x)$ as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} ranges from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b} .

Left: a region bounded by three straight lines and the graph of a function $f$. Right: approximation of the area by rectangles.

We can approximate the area of this region by drawing Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} rectangles of equal base width along the x-axis, and taking the height of each rectangle to be the height to the function graph anywhere along the extent of the rectangle's base — for example, the rightmost point. Then the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k} 'th rectangle from the left has width Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta x = (b-a)/n} and height $f(x_{k})$ where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_k = f(a + (b-a)(k/n))} and the sum of all rectangle areas is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s_n = f(x_1) \Delta x + f(x_2) \Delta x + \cdots + f(x_3) \Delta x.}

If the function is regular enough, the exact area, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s} , is given by the limit of this expression as $n$ goes to infinity,

$s=\lim _{n\to \infty }s_{n}.$

This limit is called an integral, or more technically, a Riemann integral. Its notation is the following:

$s=\int _{a}^{b}f(x)\,dx$

The equation is pronounced "$s$ equals the integral of $f$ from $a$ to $b$". It is no coincidence that the integral sign, $\scriptstyle \int$, resembles an "S" — it was originally an "S" standing for "sum", but the symbol has evolved over time. The function $f$ is called the integrand. Note the similarity between the expression $f(x)dx$ and each term in the sum for $s_{n}$: the symbol $dx$ can be understood to mean an infinitesimal width in the $x$-direction.

Calculating integrals analytically

Let us now look at the problem of calculating the integral of a function $f$ between two points $a$ and $b$. To do this, we introduce a function $F$ that gives the integral of $f$ from some fixed reference point, say $x=0$, to a point $x=t$,

$F(t)=\int _{0}^{t}f(x)\,dx.$

This function is called the primitive function of $f$. If we know the primitive function of $f$, we can calculate the integral we set out to find as

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

In words, the integral of $f$ between $a$ and $b$ is the difference between the integrals to those points, each taken from the reference point.

We usually don't have to worry about the choice of reference point, and the reason is that both terms $F(b)$ and $F(a)$ include the same contribution from the integral between the reference point and $a$, so subtracting the terms cancels that difference. An analogy is that the altitude difference between two locations on Earth is the difference between both altitudes as measured from a reference point such as sea level, but we could equally well use the center of the Earth as a reference point.

We called $F$ "the" primitive function, but every function has infinitely many primitive functions, one for each reference point. If $F(x)$ is one primitive function to $f(x)$, then so is $G(x)=F(x)+C$, where $C$ is a constant of integration that accounts for the integral of $f$ between the two reference points. Then, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G(b) - G(a) = F(b) + C - F(a) - C = F(b) - F(a)} . But except for the addition of constants, it can be shown that primitive functions are unique.

The only missing piece needed to calculate integrals is a way to actually calculate the primitive functions. To do this we need the concept of derivative, which is the rate of change of a function in a given point (the derivative of a function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is denoted by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f'\,} ). The rate of change of the primitive function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} in a point equals the value of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} in that point: the higher the graph of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is above the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} -axis, the quicker the cumulative area grows. Therefore,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F'(x) = f(x).\,}

This formula is the main result of the fundamental theorem of calculus. The fundamental theorem says that integration and differentation (the calculation of a derivative) are, essentially, inverse operations of each other. Its most immediate consequence is that if we have a table of derivatives for common functions, we can flip the columns in the table to obtain a table of primitive functions. Here is one short such table:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) [= F'(x)]}	Primitive function, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F(x)}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a\,}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ax + C\,}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ax\,}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{ax^2}{2} + C}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^n\,}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{x^{n+1}}{n+1} + C}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e^{ax}\,}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{e^{ax}}{a} + C}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{x}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ln x + C\,}

Like differentation, integration is a linear operation, so the integral of the sum of two functions, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) + g(x)} , is the sum of their primitive functions, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F(x)+G(x)} . Likewise, if we multiply a function by a constant, the primitive function is also multiplied by a constant: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c F'(x)} = Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c f(x)} .

Let us consider a concrete example:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s = \int_1^3 \left(3x^2 + \frac{1}{x}\right) dx}

The primitive function is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F(x) = x^3 + \ln x\,}

and the integral is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s = F(3) - F(1) = 26 + \ln 3.\,}

Unfortunately, if we try to calculate primitive function of complicated functions, things get more difficult. When differentiating, we can always find the derivative of a product of functions with the product rule, a quotient with the quotient rule, and a composition of functions with the chain rule. There are no general formulas of this kind for integration; instead, we have to use tricks such as integration by substitution and integration by parts. But even these tricks are not guaranteed to succeed; there are elementary functions whose primitive functions are not elementary.