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Integration
Legacy Name: Integration

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Born: October 27th, 2019

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Adopted: October 27th, 2019

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Integration:

the finding of an integral or integrals.

An integral assigns numbers to functions that describes displacement, volume, and area, among other concepts that come along with combining infinitesimal data, i.e., data that is so small there is no other way to measure it. Integration is the counterpart of differentiation in calculus.

History
Integration, where it dates back to the methods of Greek astronomer Eudoxus, had a major breakthrough in the 17th century with the onset of discovery of FTC. (See below)
While the godfathers of Calculus, Newton and Leibniz provided the approach of integration, it was formalized using limits by Riemann, thus birthing the Reimann Sum, which is an alternative approach to integration.

The intergral of the function f(x) over the range x=b to x=c gives the area under the curve between those points.

The integral with respect to x of a function of a real value os a real variable x on [a, b] can be denoted by:

If the integral goes from a finite (countable) value (a) to the upper limit of infinity, the integral expresses the limit from a to value b as b reaches infinity. If the value of integral gets closer to a finite value, the integral is to converge to that value. If this is not the case, the integral will diverge.

The properties of integrals are:
Linearity, inequalities, and conventions.

Examples of Properties:

Additive Property

Scaling by Constant

Integral of a Sum

Area Interpretation

Fundamental Theorem of Calculus

I literally have a tattoo of this irl fucking fight me.

The fundamental theorem of calculus is a statement that differentiation and integration are inverse operations.
If a continuous function is integrated and differentiated, the original function has been realized. See differentiation.