Calculus on Demand at Dartmouth College Lecture 21 | Index | Lecture 23 Lecture 22

## Resources

Math 3 Course Syllabus
Practice Exams

# 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?

### Outline

Outlines for
The Definite Integral
Properties of the Definite Integral

### Textbook

The Definite Integral
Properties of the Definite Integral

### 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/(217)
• 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) = x2 from x = 0 to x = 2, using 5 rectangles
• What is the value of the integral from –3 to 0 of √(9 – x2)?
• Simplify the integral from –a to a of sin(2x) + x.

Lecture 21 | Index | Lecture 23