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
x k = x k – x k – 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 x  4 / n. Let mk denote the midpoint of
the kth subinterval. Express the limit as an integral.
•
Revisiting Area Under a Curve
2
2
Evaluate the integral
4

x
dx
.

2
• 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.