1. AP CALC: CHAPTER 5 THE BEGINNING OF INTEGRAL FUN…

2. SECTION 5.1 – WARM UP Complete the Quick Review on page 1 of your packets

3. SECTION 5.1 - ESTIMATING WITH FINITE SUMS Objective: Estimating distance traveled, using the Rectangular Approximation Method (RAM), and Finding the volume of a sphere.

4. DISTANCE TRAVELED Explore the problem: A train moves along a track at a constant rate of 75 miles per hour from 7:00 am to 9:00 am. What is the total distance traveled by the train? Recall: distance = rate * time Graph: What shape is the graph? How do you find the area of such a shape?

5. DISTANCE TRAVELED Example1: If you travel for 3.5 hours at a constant speed of 50 mph, how far did you travel? Remember that the formula d = rt gives the total distance traveled.

6. EXAMPLE 2: A particle starts at x = 0 and moves along the x-axis with velocity v(t) = t 2 for time. Where is the particle at t = 3? Methods to approximate?

7. RECTANGULAR APPROXIMATION METHOD (RAM) NOTE: RAM is the same thing as finding Riemann Sums like we did last year (and our Michigan Maps!). We can distinguish between the 3 types (left endpoint, right endpoint, and midpoint) by the following abbreviations: LRAM (left), MRAM (midpoint), RRAM (right).

8. Example 3: A car accelerates from 0 to 88 feet per second with a speed of g(x) = -.88(x – 10) 2 + 88 feet per second after x seconds. Estimate the distance that the car travels in 8 seconds by dividing the graph into 4 sub-intervals.

9. EXAMPLE 4: The graph of y = x 2 sin x on the interval [0, 3] is crazy. Lets Graph it and then estimate the area under the curve from x = 0 to x = 3.

10. SUMMARY Sigma notation lets us express large sums (like Riemann Sums) in a compact form: When we break up a function into sub-intervals, we say that we have a partition on the interval [a, b]. The norm of a partition is the length of the partition.

13. TODAY’S AGENDA Complete the 5.2 quick review in your packets Start 5.2 notes

14. SECTION 5.2 DEFINITE INTEGRALS Riemann Sums Sigma notation lets us express large sums (like Riemann Sums) in a compact form: When we break up a function into sub-intervals, we say that we have a partition on the interval [a, b]. The norm of a partition is the length of the partition.

15. ANATOMY OF AN INTEGRAL Function Integral Start Stop Change in x

16. INTEGRATION NOTATION

17. Example 1: Express the area of the shaded region below with an integral.

18. EX. 2: The interval [-1, 3] is partitioned into n subintervals of equal length (4/n). Let denote the midpoint of the kth subinterval. Express the limit as an integral. What does 4/n say?

19. Definite Integrals and Area Example 3: Evaluate by drawing a picture

20. Example 4 Find the exact area of :.

21. EXAMPLE 5 Evaluate the integral.

22. Finding an “anti-derivative”: Power Rule :

23. Example 6

24. SECTION 5.3

25. SECTION 5.3: DEFINITE INTEGRALS AND ANTIDERIVATIVES Objectives: Using properties of definite integrals, Finding average/mean values of functions, and connecting differential and integral calculus

26. WARM UP (ON SCRAP PAPER) Calculate LRAM and RRAM using the table below: TimeVelocity 01 51.2 101.7 152.0 201.8 251.6 301.4

27. DEFINITION: For any integrable function, = (area above the x-axis) – (area below the x-axis)

28. EVALUATE THE FOLLOWING INTEGRALS WITHOUT A CALCULATOR GIVEN THAT 1. 2. 3. 4. 5. 6.7. 8.

29. THEOREM: If f(x) = c, where c is a constant, on the interval [a, b], then.

30. INTEGRALS ON A CALCULATOR!!! Evaluate the following integrals numerically on your calculators: a) b) c) Fn int (function,x,upper,lower)

31. INTEGRATING DISCONTINUOUS FUNCTIONS: Ex. 5: Use area to find. 2- (-1)= 3 units2 2 1

32. PROPERTIES OF DEFINITE INTEGRALS Rules for Definite Integrals : Order of Integration: Zero: Constant Multiple: Sum and Difference:

33. PROPERTIES OF DEFINITE INTEGRALS Rules for Definite Integrals : Additivity: + = Max – Min Inequality: SKIP Domination: on [a, b]

34. Suppose Find each f the following, if possible. 2(5)+3(7)=31 Not enough info a)b)c) 5+-2=3 d)e)f) Not enough info

35. EX. 2 (SKIP) Using Rule 6, show that the value of is less than 3/2.

36. AVERAGE VALUE OF A FUNCTION Definition: If f is integrable on [a, b], its average (mean) value on [a, b] is:

37. EX. 3: Find the average value of f(x) = 4 – x 2 on [0, 3]. Does f actually take on this value at some point in the given interval? Does function take on value? YES! At

38. DIFFERENCE BTW AVERAGE AND AVERAGE RATE OF CHANGE Average Rate of change Average of Function

39. MEAN VALUE THEOREM FOR DEFINITE INTEGRALS: If f is continuous on [a, b], then at some point c in [a, b]:

40. EXPLORATION AND EXAMPLE 4 (SKIP) Complete the exploration, “Finding the Derivative of an Integral” with your table partner!