5.2 Definite Integrals

1. 5.2 Definite Integrals • In this section we move beyond finite sums to see what happens in the limit, as the terms become infinitely small and their number infinitely large. • Sigma notation enables us to express a large sum in compact form:

2. Definite Integrals • The Greek capital letter, sigma, stands for “sum”. • The index k tells us where to begin the sum (at the number below the sigma) and where to end (at the number above the sigma). • If the symbol, infinity, appears above the sigma, it indicates that the terms go on indefinitely. • These sums are called Riemann sums. • LRAM, MRAM, and RRAM are examples of Riemann sums – not because they estimated area, but because they were constructed in a particular way.

3. Definite Integrals • Figure 5.12 is a continuous function f(x) defined on a closed integral [a , b]. • It may have negative values as well as positive values.

4. Definite Integrals • To make the notation consistent, we denote a by x0 and b by xn. The set P = {x0, x1, x2, …, xn} is called a partition of [a , b]. • The partition P determines n closed subintervals. The kth subinterval is [xk – 1 , xk], which has length xk = xk – xk – 1.

8. Definite Integrals • The value of the definite integral of a function over any particular interval depends on the function and not on the letter we choose to represent its independent variable. • If we decide to use t or u instead of x, we simply write the integral as: • No matter how we represent the integral, it is the same number, defined as a limit of Riemann sums. Since it does not matter what letter we use to run from a to b, the variable of integration is called a dummy variable.

9. Using the Notation • The interval [-1 , 3] is partitioned into n subintervals of equal length Let mk denote the midpoint of the kth subinterval. Express the limit as an integral.

11. Revisiting Area Under a Curve • Evaluate the integral

12. If an integrable function y = f(x) is nonpositive, the nonzero terms 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.

14. If an integrable function y = f(x) has both positive and negative values on an 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. • The value of the integral is the area above the x-axis minus the area below.

16. Constant Functions • Integrals of constant functions are easy to evaluate. Over a closed interval, they are simply the constant times the length of the interval (Figure 5.21).

18. Revisiting the Train Problem • A train moves along a track at a steady 75 miles per hour from 7:00 A.M. to 9:00 A.M. Express its total distance traveled as an integral. Evaluate the integral using Theorem 2.

19. Using NINT • Evaluate the following integrals numerically.

20. More Practice!!!!! • Homework – Textbook p. 282 – 283 #1 – 22 ALL.