What is Fundamental theorem of calculus?
The '''fundamental theorem of calculus''' is central to the study of [[calculus]]. It is the theorem that shows the relationship between the derivative and the integral and between the definite integral and the indefinite integral. It is broken into two parts, the first fundamental theorem of calculus and the second fundamental theorem of calculus. ==Background== A definition for [[Derivative_(mathematics)|derivative]], [[definite integral]], and [[indefinite integral]] ([[antiderivative]]) is necessary in understanding the fundamental theorem of calculus. The derivative can be thought of as measuring the change of the value of a variable with respect to another variable. The definite integral is the net area under the curve of a function and above the x-axis over an interval [a,b].<ref>“Definite integrals and negative area.” Khan Academy. 2015. June 1, 2015 <https://www.khanacademy.org/math/integral-calculus/indefinite-definite-integrals/definite_integrals/v/definite-integrals-and-negative-area></ref> The indefinite integral (antiderivative) of a function f is another function F whose derivative is equal to the first function f. ==History== The history of the fundamental theorem of calculus begins as early as the seventeenth century with [[Gottfried Wilhelm Leibniz]] and [[Isaac Newton]]. Leibniz looked at integration as the sum of infinite amounts of areas that are accumulated. As such, he references the important concept of area as it relates to the definition of the integral. Isaac Newton used [[geometry]] to describe the relationship between acceleration, velocity, and distance. This is key in understanding the relationship between the derivative and the integral; acceleration is the derivative of velocity, which is the derivative of distance, and distance is the antiderivative of velocity, which is the antiderivative of acceleration. In 1823, Cauchy defined the definite integral by the limit definition. Also in the nineteenth century, [[Siméon Denis Poisson]] described the definite integral as the d