The Fundamental Theorem of Calculus evaluate an antiderivative at the upper and lower limits of integration and take the difference. This theorem is separated into two parts. The first part is called the first fundamental theorem of calculus and states that one of the antiderivatives of some function may be obtained as the integral of the function with a variable bound of integration. The second part of the theorem, called the second fundamental theorem of calculus, states that the integral of a function over some interval can be computed by using any one of its infinitely many antiderivatives.

The fundamental theorem of calculus along curves states that if has a continuous infinite integral in a region containing

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The mathematician who discovered what we call the fundamental theorem of calculus is Isaac Newton. For him, this theorem was virtually self-evident, as Victor Katz states on the book “A History of Mathematics”.

Isaac Newton was born in England on January 4,1643 and is a known as physicist mathematician, astronomer, natural philosopher, alchemist and theologian. He is considered as one of the most influent men in history by developing the principles of modern physics.

On 1661, Newton entered University of Cambridge’s Trinity College in a program similar to a work-study program, as his uncle was pushing him to do so. He spent all his free time reading from the modern philosopher and the result was amazing.

On 1665, Newton approached mathematics with a new perspective: infinitesimal calculus. He started developing the theory of calculus that Englishmen John Wallis and Isaac Barrow was working on earlier. Thus, calculus allowed mathematicians and engineers to make sense of the motion and dynamic change in the changing world around us. Even though he had to follow a tough path he was able to publish Philosophiae Naturalis Principia Mathematica (Principia) in 1687.This book contains information on all of the essential concepts of physics, except

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On 1705, Newton’s career was known by everyone and he was knighted by Queen Anne of England.

He died on March 31, 1727. Back to the Fundamental Theorem of Calculus. This theorem is a simple theorem that establishes the relationship between the derivative and the integral and as we said before has two parts. The first part that talks about the relationship between the derivative and the integrals and the second part that tells us how to calculate a definite integral.

The easiest way to explain how this theorem works is:

• The First Fundamental Theorem of Calculus:

First, let’s say we have a function f(x)

Now

We take two points on the x axis: a and x. And we consider the area under the curve from a to x:

If we take a smaller , we will get a smaller area:

And if we take a greater , then we’ll have a bigger area:

So we can define our area function as A(x)->(depends on what curve we’re using)

We know that the area under the graph of a function f(t) from a to x is: and our function will be:

So the first part of the fundamental theorem of calculus