The area of a rectangle between x lo and x hi, where x hi > x lo, is the definite integral of 1. Except for the uncertainty of an additive real constant, C, the indefinite integral of 1 is not specified in general. In the particular case where x lo = 0, however, the indefinite integral of 1 equals x hi. Therefore, if we let C be any number other than zero, then the indefinite integral of 1 over x from 0 to 1 is equal to 1 + C.

In mathematics, integration is the operation that results in the value of a function at a point or over a range of points. The adjective integrative refers to **a measurement device** used for measuring the amount of light transmitted through or reflected by a sample; it is thus able to provide **a quantitative estimate** of the concentration of substances in the sample. The word "integral" comes from the Latin integrale, meaning "whole," "entire." Thus, integration is the process of adding up the values of a function at **separate points**.

Integration can also be defined as the action of determining the area under a curve. This is equivalent to integrating the function from 0 to t with respect to its variable t. Integration with respect to another variable (such as x in this example) is called antiderivative evaluation.

One of the two major ideas in mathematics is integration, and the integral assigns a number to the function. There are two kinds of integrals: definite integrals and indefinite integrals. A definite integral is calculated by taking a sum of multiple copies of a function, dividing the sum by the number of copies used and rounding down to the nearest whole number. For example, if I say that f(x) = x^3, then the area under the curve from 0 to 1 is equal to **1/2 times the value** of the integral from zero to one of x^3. Indefinite integrals are not calculated by multiplying a number by an expression; instead they are calculated by solving an equation for **a particular value** of the parameter involved. For example, if I say that g(x) = x - e^x, then the area under the curve from 0 to 1 is equal to 1/2 times the value of the integral from zero to infinity of e^x minus 1.

There are several ways to integrate a function. The most common method is by parts.

If this limit exists, define **definite integrals**. The integrand is the function f (x), and the variable x is the variable of integration. The integers a and b are referred to as the limits of integration, with a being the lower limit of integration and b being the upper limit of integration. Then integration is the process of computing the area under the curve defined by f (x) from a to **b. Integration** is the opposite of differentiation. As the limit of the number of times you can differentiate **any given function** is infinite, the limit of the number of times you can integrate it is also infinite.

Integration is the process of adding up all the small pieces of something whole. For example, if you had a pile of sand that was getting smaller and smaller, then you would say that the volume of this pile is decreasing, even though there are an infinite number of grains in the pile. In the same way, if you were to integrate the equation y = x, then the result would be infinity because there are an infinite number of values for x on the real number line.

Integration is useful for calculating areas under a curve. If you have a flat surface with no holes, then the area beneath the curve f(x) from a to b is equal to the integral from a to b of f(x). In other words, integration is the process of calculating a flat surface's area.