Understanding Definite Integrals and Riemann Sums

1. Ch 5.2 Definite Integrals

2. Ch 5.2 Introduction • In 5.1 (the last section) we used MRAM, LRAM, and RRAM to approximate area. • You will hear these referred to as Reimann Sums (named after Georg Freidrich Riemann) • In 5.2 we find exact area, using “The Definite Integral”

3. Review of Sigma Notationfrom PreCalc/ALG 2 Sigma Notation enables us to express a large sum in compact form Where to end Where to begin

4. Revisiting EX 1 from 5.1 A particle starts at x = 0 and moves along the x-axis with velocity v(t) = t² for t ≥ 0. When is the particle at t = 6? We had set up a midpoint approximation with 3 intervals of length 2. We could write the following: (2)

5. What happens as the number of rectangles increases?We get a more and more accurate approximation of the area under the curve.

6. *The Definite Integral is a Limit of Riemann Sums* Let f be a continuous function defined on a closed interval [a, b], partitioned into n subintervals of equal length Δx = (b-a)/n. Let the numbers be chosen arbitrarily in the kth interval. Then: The function is the integrand “Integral Sign” Upper limit of integration Differential (the variable of integration) Lower limit of integration *When you find the value of the integral you have evaluated the integral

7. EX 2: Using the Notation The interval [-1,3] is partitioned into n subintervals of equal length Δx. Let denote the midpoint of the kth subinterval. Express the limit As an integral.

8. You Try! #2-6 even on your classwork

9. THM 1: The Existence of Definite Integrals All continuous functions are integrable. That is, if a function f is continuous on an interval [a,b], then its definite integral over [a,b] exists.

10. How do I evaluate an “Integral”? In this chapter you will do it two different ways depending on the problem: • By the graph:calculating area based on geometric shapes which you already know the area for – but be careful with negative velocity. If f(x) ≤ 0, • By the antiderivative = = f(b) – f(a) a) b) EX 3: Evaluating with both methods

11. What if the entire curve is below the x-axis? DEF PG 279: If an integrable function y = f(x) is nonpositive, the nonzero terms f()Δx in the Riemann sums for f over an interval [a, b] are negatives of rectangle areas. The limit of the sums, the integral of f from a to b, is therefore the negative of the area of the region between the graph of f and the x-axis.

12. What if the function y = f(x) has both positive and negative values on the interval [a, b]? Then the Riemann sums for f on [a, b] add areas of rectangles that lie above the x-axis to the negatives of areas of rectangles that lie below the x-axis. ***The resulting cancellations mean that the limiting value is a number whose magnitude is less than the total area between the curve and the x-axis. This is why the integral is sometimes called referred to as net area of the region ***

13. You Try! #8-12 even on your Classwork THM 2: The Integral of a Constant If f(x) = c where c is a constant, on the interval [a, b], then:

14. EX 4: Evaluating an integral with area

15. You Try! classwork # 14-22evenEVALUTE WITH AREAAAAA

16. Properties of Integrals • Order of integration Rule: = - • Constant Multiple Rule: = k • Sum and Difference Rule: =

17. Exploration 1 1 -2 0 2pi + 2 2 4 0 0 4

18. Finish the classwork! • 5.2 In-Class Groupwork:pg 282 EX #2-28 even, 34, 40 • 5.2 HW: pg 283 EX #13-27odd, 33, 35, 39,