1. Riemann Sums and the Definite Integral Lesson 5.3

2. Why? Why is the area of the yellow rectangle at the end = a b

3. Review We partition the interval into n sub-intervals Evaluate f(x) at right endpoints of k th sub-interval for k = 1, 2, 3, … n a b f(x)

4. Review Sum We expect S n to improve thus we define A, the area under the curve, to equal the above limit. a b f(x) Look at Goegebra demo Look at Goegebra demo

5. Riemann Sum 1.Partition the interval [a,b] into n subintervals a = x 0 < x 1 … < x n-1 < x n = b Call this partition P The k th subinterval is  x k = x k-1 – x k Largest  x k is called the norm, called ||P|| 2.Choose an arbitrary value from each subinterval, call it

6. Riemann Sum 3.Form the sum This is the Riemann sum associated with the function f the given partition P the chosen subinterval representatives We will express a variety of quantities in terms of the Riemann sum

7. The Riemann Sum Calculated Consider the function 2x 2 – 7x + 5 Use  x = 0.1 Let the = left edge of each subinterval Note the sum

8. The Riemann Sum We have summed a series of boxes If the  x were smaller, we would have gotten a better approximation f(x) = 2x 2 – 7x + 5

9. The Definite Integral The definite integral is the limit of the Riemann sum We say that f is integrable when  the number I can be approximated as accurate as needed by making ||P|| sufficiently small  f must exist on [a,b] and the Riemann sum must exist

10. Example Try Use summation on calculator.

11. Example Note increased accuracy with smaller  x

12. Limit of the Riemann Sum The definite integral is the limit of the Riemann sum.

13. Properties of Definite Integral Integral of a sum = sum of integrals Factor out a constant Dominance

14. Properties of Definite Integral Subdivision rule a b c f(x)

15. Area As An Integral The area under the curve on the interval [a,b] a c f(x) A

16. Distance As An Integral Given that v(t) = the velocity function with respect to time: Then Distance traveled can be determined by a definite integral Think of a summation for many small time slices of distance

17. Assignment Section 5.3 Page 314 Problems: 3 – 47 odd