The Fundamental Theorem of Calculus

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Taught by Houston
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Lesson Summary:

In this lesson on the Fundamental Theorem of Calculus, we learn about the terminology and symbolism involved in integrals, such as the integrand and differential. We also explore the Average Value Theorem, which states that there exists a number c in the open interval a, b, such that f of c is equal to the average value of f over the interval. Finally, we delve into the derivative of phi, which is the function defined by integration, and how it relates to the Fundamental Theorem of Calculus. Examples are provided to illustrate these concepts.

Lesson Description:

Average value theorem. The function p(x) = integral of f(s) ds. The fundamental theorem of calculus.

Copyright 2005, Department of Mathematics, University of Houston. Created by Selwyn Hollis. Find more information on videos, resources, and lessons at

Additional Resources:
Questions answered by this video:
  • What is the Fundamental Theorem of Calculus?
  • What is a dummy variable?
  • What is the average value of a function on an interval?
  • What is the minimum and maximum value of a function on an interval?
  • What is the Average Value Theorem?
  • What is the Mean values theorem for integrals?
  • What is an antiderivative?
  • How do you compute an integral?
  • What is antidifferentiation?
  • How do you find the area bounded by a graph?
  • Staff Review

    • Currently 5.0/5 Stars.
    This video starts off with a review of some terminology and symbolism in Calculus, such as the integral sign, lower and upper limits, the integrand, and the differential. Some theorems and results are discussed, and finally we get to the Fundamental Theorem of Calculus. Then, some variations of the Fundamental Theorem are discussed. Some examples are also computed.