The Fundamental Theorem of Calculus

Read this section to see the connection between derivatives and integrals. Work through practice problems 1-5.

The Fundamental Theorem of Calculus

This section contains the most important and most used theorem of calculus, THE Fundamental Theorem of Calculus. Discovered independently by Newton and Leibniz in the late 1600s, it establishes the connection between derivatives and integrals, provides a way of easily calculating many integrals, and was a key step in the development of modern mathematics to support the rise of science and technology. Calculus is one of the most significant intellectual structures in the history of human thought, and the Fundamental Theorem of Calculus is a most important brick in that beautiful structure.

The previous sections emphasized the meaning of the definite integral, defined it, and began to explore some of its applications and properties. In this section, the emphasis is on the Fundamental Theorem of Calculus. You will use this theorem often in later sections.

There are two parts of the Fundamental Theorem. They are similar to results in the last section but more general. Part 1 of the Fundamental Theorem of Calculus says that every continuous function has an antiderivative and shows how to differentiate a function defined as an integral. Part 2 shows how to evaluate the definite integral of any function if we know an antiderivative of that function.

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