Integral

For non-mathematical meanings of "Integral", see integration (non-mathematical).

It is recommended that the reader be familiar with algebra, derivatives, functions, and limits.

In mathematics, the term "integral" has two unrelated meanings; one relating to integers, the other relating to integral calculus.

"Integral" in relation to integers

A real number is "integral" if it is an integer. The integral value of a real number x is defined as the largest integer which is less than, or equal to, x. The integral value of x is often denoted by ⌊x⌋ and called the "floor function".

In abstract algebra, an integral domain is a commutative ring with 0 ≠ 1 in which the product of any two non-zero elements is always non-zero. Integral domains are generalizations of the integers.

Integral Calculus

In calculus, the integral, of a function, can be used to represent areas, masses, volumes, totals, or averages. There are several technical definitions of integral which make this notion more precise. All technical definitions of integral generalize the calculation of area and volume for regular figures and solids. In particular, for a constant function, the integral is defined as its constant value times the measure of the region on which it is defined; in this basic case, the integral is just the area of a rectangle (in one dimension) or volume of a prism or cylinder (in two dimensions). The integral of a general function is then defined as the limit of the easily-calculated integrals of a sequence of simpler functions.

If a function has an integral, it is said to be integrable. The function for which the integral is calculated is called the integrand. If the region over which the integral is calculated is fixed, the integral is a definite integral. If the region is variable, the integral is an indefinite integral. In one dimension, a region may be an interval. If so, the greatest lower bound of the interval is called the lower limit of integration and the least upper bound is called the upper limit of integration.

Area under the curve in one dimension

Consider a real function f of one real variable x. Its integral is the size of the area bounded by the x-axis and the graph of a function, f(x); negative areas are possible. Integrals are calculated by integration, which is a so-called "accumulation process" (see below).

Integral defined as area under a curve

Let f(x) be a function of the interval [a,b] into the real numbers. For simplicity, assume that this function is non-negative (it takes no negative values.) The set S=Sf:={(x,y)|0≤yf(x)} is the region of the plane between f and the x axis. Measuring the "area" of S is desirable, and this area is denoted by ∫f, and it is the (definite) integral of f.

Improper and Trigonometric Integrals

If either the interval of integration, or the range of the function, is infinite; the integral is an "improper integral". Integrals which involve trigonometric functions, are trigonometric integrals. Some integrals can be evaluated via trigonometric substitution.

Means of Integration

The following pages discuss means of integrating various functions:

Riemann and Lebesgue Integrals

One should examine the articles on Riemann and Lebesgue integrals. The concept of Riemann integration was developed first, and Lebesgue integrals were developed to deal with pathological cases for which the Riemann integral was not defined. If a function is Riemann integrable, then it is also Lebesgue integrable, and the two integrals coincide.

The antiderivative approach occurs when we seek to find a function F(x) whose derivative F(x) is some given function f(x). This approach is motivated by calculus, and is the main method used for calculating the area under the curve as described in the preceding paragraph, for functions given by formulae.

Functions which have antiderivatives are also Riemann integrable (and hence Lebesgue integrable.) The nonobvious theorem that states that the two approaches ("area under the curve" and "antiderivative") are in some sense the same as the fundamental theorem of calculus

(And the relationships works in reverse; the Radon-Nikodym derivative can be pulled out of the measure machinery underlying Lebesgue integrals.)

The nuance between Riemann and Lebesgue integration

Both the Riemann and the Lebesgue integral are approaches to integration which seek to measure the area under the curve, and the overall schema in both cases is the same.

First, we select a family of elementary functions, for which we have an obvious way of measuring the area under the curve. In the case of the Riemann integral, this choice is so that the area under the curve can be regarded as a finite union of rectangles, and the functions are called step functions. For the Lebesgue integral, "rectangle" is replaced by something more sophisticated, and the resulting functions are called simple functions.

Then we try to impose monotonicity. If 0≤fg (and hence Sf is a subset of Sg) then we should have that ∫f≤∫g. With this monotonicity requirement, for an arbitrary non-negative function f, we can approximate its area from below using a carefully chosen elementary function s (in the case of Riemann integration, a step function, and in the case of Lebesgue integration, a simple function.) We choose s so that sf but s is very close to f. The area under s is a lower bound for the integral of f, and it is called a lower sum. In the case of the Riemann integral, we also produce upper sums in a similar fashion: we choose step functions, say s, so that sf but s is very close to f, and we regard such an upper sum as an upper bound for the area under f. The Lebesgue theory does not use upper sums.

Lastly, a limit-taking step is taken to make the elementary functions approach f more and more closely, and an area is obtained for some functions f. The functions which we can integrate are said to be integrable. However, the differences begin here; the Riemann theory was simpler thus far, but its simplicity results in a more limited set of integrable functions than the Lebesgue theory. In addition, the interaction between limits and the integral are more difficult to describe in the Riemann setting.

Other integrals

Although the Riemann and Lebesgue integrals are the most important ones, a number of others exist, including but not limited to: