[[Isaac Newton]] was one of the most famous inventors of calculus, however there are several others who developed calculus simultaneously including [[Gottfried Leibniz]].<ref>Calculus, 5th edition, by James Stewart, Nelson Canada</ref> The [[Calculus Controversy|disagreement]] over the originator of calculus created rifts within the European mathematical community for years.
==Integrals==
{{Main|Integral}}
Integration is primarily defined as the method to calculate the area in an xy-plane above the x-axis. There are two fundamentally different kinds of integrals.
===Indefinite Integrals===
{{main|Indefinite integral}}
The antiderivative of a function is often called the ''indefinite integral''. (Indefinite because the limits a and b haven't been specified.) So, for example, the derivative of <math>\frac{x^3}{3}+7</math> is <math>x^2</math>. From this it follows that the antiderivative of
<math>x^2</math> could be <math>\frac{x^3}{3}+7</math>. But note that the "7" in that formula was a red herring. Adding any constant to a function doesn't change its derivative, so the antiderivative of <math>x^2</math> could have any constant added to it. This arbitrary constant is usually written '''C''' and is called the "constant of integration". The indefinite integral could be written:
:<math>\int x^2\ \mathrm{d}x = \frac{x^3}{3} + C\,</math>,
The Fundamental Theorem of Calculus says that the area under the graph of <math>x^2</math> between a and b is the difference in the values of <math>\frac{x^3}{3}+C</math> between a and b. Note that the constant of integration cancels out.
===Definite Integration===
This kind of integral is called a ''definite integral'', written with the limits:
:<math>\int_a^b x^2\ \mathrm{d}x = \frac{b^3}{3} - \frac{a^3}{3}\,</math>,
The above is a simplified "intuitive" treatment of calculus and of this theorem. The actual "rigorous" proof, "rigorous" definitions of derivative and integral, and statement of the conditions under which the theorem is true, are beyond the scope of this article.