1. The Fundamental Theorem of Calculus We’ve learned two different branches of calculus so far: differentiation and integration. Finding slopes of tangent lines and finding areas under curves seem unrelated, but in fact, they are very closely related. It was Isaac Newton’s teacher at Cambridge University, a man name Isaac Barrow (1630 – 1677), who discovered that these two processes are actually inverse operations of each other in much the same way division and multiplication are. It was Newton and Leibniz who exploited this idea and developed the calculus into it current form. The Theorem Barrow discovered that states this inverse relation between differentiation and integration is called The Fundamental Theorem of Calculus.

2. Fundamental Theorem of Calculus 1 (FToC1) The integral gives the NET change Fundamental Theorem of Calculus 2 (FToC2)

3. This result, while taught early in elementary calculus courses, is actually a very deep result connecting the purely algebraic indefinite integral and the purely analytic (or geometric) definite integral. FToC1 FToC1 bridges the antiderivative concept with the area problem. To evaluate an integral, take the antiderivatives and subtract. It can be proved and the proof can be found elsewhere (WEB) It is a “shortcut” rule for integration: an easier way (from Riemann sums and other methods) to calculate definite integrals.

11. Lower limit of integration is a constant. Or, longer way: Directly:

13. = = The upper limit of integration does not match the derivative, but we could use the chain rule. = == The lower limit of integration is not a constant, but the upper limit is. We can change the sign of the integral and reverse the limits. What if upper limit is g(x) not x itself? Or upper limit is constant and lower limit is x?

14. FToC 2, the most general form

15. Example 10: Evaluate the following using the FTOC2, then if feeling bored verify by doing in the Loooooooong way.

16. The Mean Value Theorem (for Integrals)

17. 1

20. The average rate of change of a car’s position over an interval is represented graphically as the slope of the secant line to the graph position/time over the interval. Average value of velocity over the same interval is:

21. Area trapezoidal approximation is more accurate that LRAM (basically arithmetic mean)

22. Sometimes we might have to solve an integral equation! Being able to simplify definite integrals with variables in the interval of integration is important. Here are a couple of examples showing an important application that is important.

25. In the previous example, the first method relied heavily upon our ability to find the antiderivative of the integrand. This is not always easy, possible, or prudent! Being able to express a particular value of a particular solution to a derivative as a definite integral is of paramount importance, especially when we don’t know how to find a general antiderivative. (calculator can do easily definite integrals – see problem 20) Hard Facts To Refute: A. Where you are at any given time is a function of 1) where you started and 2) where you’ve gone from your starting point (displacement). B. What you have at any given moment is a function of 1) what you started with plus 2) what you’ve accumulated since then. When you accumulate at a variable rate, you can to use the definite integral to find your net accumulation. Important Idea of Accumulation***************************(* means VERY IMPORTANT) What one has now = What one started with + What one has accumulated since one started. This can be expressed mathematically as

26. Example 19: (a) and integral equation for f(x) (b) f(3) (c) f(-2)

27. Example 19 continues: