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:

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.

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.

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.

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.

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.

4 / .x n

Revisiting Area Under a Curve• Evaluate the integral

2 2

24 .x dx

• 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.

• 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.

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).

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.

Using NINT• Evaluate the following integrals numerically.

More Practice!!!!!

• Homework – Textbook p. 282 – 283 #1 – 22 ALL.