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: