Warm-Up: January 31, 2013

1. Warm-Up: January 31, 2013 • Use your RIEMANN program to find the area under f(x) from x=a to x=b using each rectangular approximation method (LRAM, MRAM, and RRAM) with n=10

2. Homework Questions?

3. Definite Integrals Section 5.2

4. Sigma Notation • The Greek capital Sigma, Σ, stands for “sum.” • The index, k, tells us where to start (below Σ) and where to end (above Σ).

5. Riemann Sums • RRAM, MRAM, and LRAM are examples of Riemann sums • Function f(x) on interval [a, b] • We partitioned the interval into nsubintervals (to make the bases of our n rectangles). • Let’s name the x-coordinates at the corners of our rectangles

6. Riemann Sums • Let’s denote a by x0 and b by xn • The corners of our rectangles make the set • P is called a partition of [a,b] • Our subintervals are the closed intervals • The width of the kth subinterval has width

7. Riemann Sums • We choose some x value inside each subinterval to be the height of the rectangle. • RRAM: Use the right endpoint • MRAM: Use the midpoint • LRAM: Use the left endpoint • Could use any other point • Let’s call the x value from the kth subinterval ck

8. Riemann Sums • The area of each rectangle is the product of its height, f(ck), and its width, Δxk • The sum of these areas is • This is the Riemann Sum for f on the interval [a, b]

9. Question • Have we done any calculus yet?

10. The Definite Integral • I is the definite integral of f over [a, b] • ||P|| is the longest subinterval length, called the norm of the partition

11. Definite Integrals ofContinuous Functions • All continuous functions are integrable. • Let f be continuous on [a, b] • Let [a, b] be partitioned into n subintervals of equal length • The definite integral is given by • where each ck is chosen arbitrarily in the kth subinterval

12. Integration Notation

13. Integration Notation The function is the integrand Upper limit of integration x is the variable of integration Integral sign Lower limit of integration Integral of f from a to b

14. Assignment • Read Section 5.2 (pages 258-266) • Seriously, read Section 5.2

15. Warm-Up: February 1, 2013 • Use your RIEMANN program to estimate

16. Homework Questions?

17. Example 1 • Express the limit as a definite integral

18. You-Try #1 • Express the limit as a definite integral

19. Area Under a Curve • If y=f(x) is nonnegative and integrable on [a, b] • If y=f(x) is nonpositive and integrable on [a, b]

20. Integrals and Areas • For any integrable function:

21. Evaluating Integrals • We can use geometric areas to evaluate certain integrals, including: • Constant functions (rectangles) • Linear functions (trapezoid) • Semi-circles

22. Example 2

23. Integrals of Constants • If f(x)=c, where c is a constant, then:

24. Example 3

25. Integrals on TI-83 • [MATH] [9:fnInt] • fnInt(function, X, lower bound, upper bound) • or • Enter function into Y1 • [2nd] [TRACE] [7:∫f(x)dx] • Enter lower limit and upper limit

26. You-Try #4 • Approximate the following to three decimal places:

27. Assignment • Read Section 5.2 (pages 258-266) • Page 267 Exercises 1-27 odd • Read Section 5.3 (pages 268-274)