the Fundamental Theorem of Calculus and the Mean Value Theorem for Integrals

In the previous post, we have already worked with the FTC Part I or the First FTC. I include it again in this section with some additional problems to practice from Larson 7 Ch 4 Section 4.





Related to this is the Mean Value Theorem for Integrals and the Average Value of a Function. You can see that the two are just slight variations of each other. In both f(c) refers to the average value of f(x) on the closed interval [a,b].









Then we have the FTC Part 2 or the Second FTC. Once you get the idea it is fairly straight forward. When the upper limit of integration is a function of x and not just x, remember to apply the chain rule and multiply by the derivative of the upper limit.





And finally a few problems to put it all together:

Definite Integrals

There are two special properties of the definite integral that are explained in the following graphic. Examples are also provided.



Below are exercises related to solving definite integrals algebraically. This formula is known as the Fundamental Theorem of Calculus Part I. Guidelines for evaluating definite integrals using the FTC Part I are below:





Below are exercises that utilize the three different RAM methods.



Below are exercises that utilize what is called the trapezoidal approximation method or the trapezoidal rule. The definition of the trapezoidal rule is shown below:



And then an example of the trapezoidal rule in action:



And an assignment:

Definite Integrals - Using Known Geometric Formulas

We started on these today in class. Please complete the remaining problems for hw tonight. :)