What is the Fundamental Theorem of Calculus? 

During the development of mathematical theory, scientists began to develop new techniques to solve problems. These methods allowed them to explain many phenomena. They could even calculate three-dimensional motions of objects and measure distances in space. One of the most important theoretical achievements was the Fundamental Theorem of Calculus. It established a relationship between differentiating and integrating functions. As a result, students were able to learn how to evaluate definite integrals. Moreover, it gave them a method for evaluating definite integrals without the use of Riemann sums. 

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This fundamental theorem establishes a relationship between the differentiation and the integration of a function. It also establishes the concept of indefinite and definite integration. Specifically, it connects the algebraic indefinite integral and the analytic definite integral. Essentially, it states that the difference between the integrals of a function is equal to the rate of change of its area under a curve. 

The first Fundamental Theorem of Calculus was proved by James Gregory. Later, Isaac Barrow proved a more general version of this theorem. His more generalized version has become the second Fundamental Theorem of Calculus. In this theorem, a definite integral is defined over a closed interval. The theorem is sometimes called the Newton-Leibniz axiom. Among other things, it explains how the derivative of an integrable function can be evaluated. 

Using Fundamental Theorem of Calculus, part 2, you can find the area under a curve. If you know the rate of change of an area under a curve, you can find its height. You can also use your calculator to estimate this area. 

Another interesting point about the Fundamental Theorem of Calculus is that it relates a definite and an indefinite integral. Moreover, it is also a practical tool for computation. With this theorem, you can find out the area under a curve easily. By using Fundamental Theorem of Calculus, Part 2, you can also calculate the definite integral of a function. 

Depending on your starting point, you can get a constant value. Therefore, the F-value theorem tells you how to calculate the definite integral by evaluating the F-values. Similarly, you can also use the symbolic integration to calculate the antiderivative. A semidefinite integral is a different type of integral. However, the fundamental theorem of calculus still says that a continuous function with a definite derivative has a definite antiderivative. 

The second Fundamental Theorem of Calculus is often referred to as the Newton-Leibniz axiom. As with the first, this theorem states that a definite integral is a function that is the value of an antiderivative of a function at a given limit. Unlike the first theorem, it does not assume continuity over the entire interval. 

When calculating a definite integral with the Fundamental Theorem of Calculus, you can use the area function to estimate the height of the area under a curve. The area function is the derivative of f(x). Moreover, you can use the area function to calculate the cumulative effect of small contributions.