1. Chapter 6 The Definite Integral

2. § 6.1 Antidifferentiation

3. DefinitionExample Antidifferentiation: The process of determining f (x) given f ΄(x) If, then

4. Finding AntiderivativesEXAMPLE Find all antiderivatives of the given function.

5. Theorems of Antidifferentiation

6. The Indefinite Integral

7. Rules of Integration

8. Finding AntiderivativesEXAMPLE Determine the following.

9. Finding AntiderivativesEXAMPLE Find the function f (x) for which and f (1) = 3.

10. Antiderivatives in ApplicationEXAMPLE A rock is dropped from the top of a 400-foot cliff. Its velocity at time t seconds is v(t) = -32t feet per second. (a) Find s(t), the height of the rock above the ground at time t. (b) How long will the rock take to reach the ground? (c) What will be its velocity when it hits the ground?

11. § 6.2 Areas and Riemann Sums

12. Area Under a Graph DefinitionExample Area Under the Graph of f (x) from a to b: An example of this is shown to the right

13. Area Under a Graph In this section we will learn to estimate the area under the graph of f (x) from x = a to x = b by dividing up the interval into partitions (or subintervals), each one having width where n = the number of partitions that will be constructed. In the example below, n = 4. A Riemann Sum is the sum of the areas of the rectangles generated above.

14. Riemann Sums to Approximate AreasEXAMPLE SOLUTION Use a Riemann sum to approximate the area under the graph f (x) on the given interval using midpoints of the subintervals The partition of -2 ≤ x ≤ 2 with n = 4 is shown below. The length of each subinterval is -22 x1x1 x2x2 x3x3 x4x4

15. Riemann Sums to Approximate Areas Observe the first midpoint is units from the left endpoint, and the midpoints themselves are units apart. The first midpoint is x 1 = -2 + = -2 +.5 = -1.5. Subsequent midpoints are found by successively adding CONTINUED midpoints: -1.5, -0.5, 0.5, 1.5 The corresponding estimate for the area under the graph of f (x) is So, we estimate the area to be 5 (square units).

16. Approximating Area With Midpoints of IntervalsCONTINUED

17. Riemann Sums to Approximate AreasEXAMPLE SOLUTION Use a Riemann sum to approximate the area under the graph f (x) on the given interval using left endpoints of the subintervals The partition of 1 ≤ x ≤ 3 with n = 5 is shown below. The length of each subinterval is 3 x1x1 x2x2 x3x3 x4x4 x5x5 11.41.82.22.6

18. Riemann Sums to Approximate Areas The corresponding Riemann sum is CONTINUED So, we estimate the area to be 15.12 (square units).

19. Approximating Area Using Left EndpointsCONTINUED

20. § 6.3 Definite Integrals and the Fundamental Theorem

21. The Definite Integral Δx = (b – a)/n, x 1, x 2, …., x n are selected points from a partition [a, b].

22. Calculating Definite IntegralsEXAMPLE SOLUTION Calculate the following integral. The figure shows the graph of the function f (x) = x + 0.5. Since f (x) is nonnegative for 0 ≤ x ≤ 1, the definite integral of f (x) equals the area of the shaded region in the figure below. 1 0.5 1

23. Calculating Definite Integrals The region consists of a rectangle and a triangle. By geometry, CONTINUED Thus the area under the graph is 0.5 + 0.5 = 1, and hence

24. The Definite Integral

25. Calculating Definite IntegralsEXAMPLE Calculate the following integral.

26. The Fundamental Theorem of Calculus

27. EXAMPLE Use the Fundamental Theorem of Calculus to calculate the following integral. Use TI 83 to compute the definite integral: 1) put f(x) into y1 and graph. 2) 2 nd trace 7 3) Enter lower limit and upper limit at the prompts.

28. Area Under a Curve as an Antiderivative

29. § 6.4 Areas in the xy-Plane

30. Properties of Definite Integrals

31. Area Between Two Curves

32. Finding the Area Between Two CurvesEXAMPLE Find the area of the region between y = x 2 – 3x and the x-axis (y = 0) from x = 0 to x = 4.

33. Finding the Area Between Two CurvesEXAMPLE Write down a definite integral or sum of definite integrals that gives the area of the shaded portion of the figure.

34. § 6.5 Applications of the Definite Integral

35. Average Value of a Function Over an Interval

36. EXAMPLE Determine the average value of f (x) = 1 – x over the interval -1 ≤ x ≤ 1.

37. Average Value of a Function Over an IntervalEXAMPLE (Average Temperature) During a certain 12-hour period the temperature at time t (measured in hours from the start of the period) was degrees. What was the average temperature during that period?

38. Consumers’ Surplus

39. EXAMPLE Find the consumers’ surplus for the following demand curve at the given sales level x.