Additional Resources
OpenStax Online Textbook
How to Read a Math Textbook
Additional Instructional Resources
The Fundamental Theorem of Calculus- FTC5
Objectives
- Describe the meaning of the Mean Value Theorem for Integrals.
- State the meaning of the Fundamental Theorem of Calculus, Part 1.
- Use the Fundamental Theorem of Calculus, Part 1, to evaluate derivatives of integrals.
- State the meaning of the Fundamental Theorem of Calculus, Part 2.
- Use the Fundamental Theorem of Calculus, Part 2, to evaluate definite integrals.
- Explain the relationship between differentiation and integration.
Summary
We can find the exact value of a definite integral without taking the limit of a Riemann sum or using a familiar area formula by finding the antiderivative of the integrand, and hence applying the Fundamental Theorem of Calculus.
The Fundamental Theorem of Calculus says that if is a continuous function on and is an antiderivative of then
Hence, if we can find an antiderivative for the integrand evaluating the definite integral comes from simply computing the change in on
See the Desmos Demonstration on Evaluating Definite Integrals.
A slightly different perspective on the FTC allows us to restate it as the Total Change Theorem, which says that
for any continuously differentiable function This means that the definite integral of the instantaneous rate of change of a function on an interval is equal to the total change in the function on