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


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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

  • click to see the video3 + 5 + 7 + 9 + 11 = summation from i = 0 to 4 of (3 + 2i)
  • click to see the video– 1/2 + 2/4 – 3/8 + 4/16 – 5/32 + ... + – 17/(217)
  • click to see the videoApproximate area under a curve by adding areas of rectangles
  • click to see the videoWhat integral equals the limit as n approaches ∞ of the summation from i = 0 to n of (1 + (2i/n)2) · 2/n ?
  • click to see the videoEstimate area under f(x) = x2 from x = 0 to x = 2, using 5 rectangles
  • click to see the videoWhat is the value of the integral from –3 to 0 of √(9 – x2)?
  • click to see the video Simplify the integral from –a to a of sin(2x) + x.

Lecture 21 | Index | Lecture 23