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Contents
In this lecture we define the definite integral and develop some of its immediate properties.
Quick Question
What is the total area under the sum of the two functions f(x) = 1 and g(x) = 1/2 + x/2?
Answer
Outline
Outlines for
The Definite Integral
Properties of the Definite Integral
Textbook
The Definite Integral
Properties of the Definite Integral
Today's Homework
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Quiz
The Definite Integral Quiz
Properties of the Definite Integral Quiz
Examples
 Write the following sum in sigma notation: 5/n + 50/n + 375/n + ... + (n − 1) 5^{(n − 1)}/n.
 Consider the function f(x) defined to be 1 if x is in the interval [0, 1] and is rational, and 0 otherwise. Is it possible to find the area between this function and the x axis?
 Calculate the integral from 0 to 1 of x using the limits of Riemann sums.
 Evaluate the integral from −3 to 6 of x.
 Write a summation as a definite integral.
 Evaluate the integral from 0 to 2π of sin(x) + 5x.
Applets

Riemann Sums

Mean Value Theorem for Integrals
Videos
 3 + 5 + 7 + 9 + 11 = summation from i = 0 to 4 of (3 + 2i)
 – 1/2 + 2/4 – 3/8 + 4/16 – 5/32 + ... + – 17/(2^{17})
 Approximate area under a curve by adding areas of rectangles
 What integral equals the limit as n approaches ∞ of the summation from i = 0 to n of (1 + (2i/n)^{2}) · 2/n ?
 Estimate area under f(x) = x^{2} from x = 0 to x = 2, using 5 rectangles
 What is the value of the integral from –3 to 0 of √(9 – x^{2})?

Simplify the integral from –a to a of sin(2x) + x.
