MATH 1712: WEEK 9

Topics for this week are:

• Integration by Substitution
• Areas and the Definite Integral
• The Fundamental Theorem of Calculus; Properties of the Definite Integral

Objectives for this week:

• Use u-substitution to evaluate definite and indefinite integrals
• Find the area under a curves using Riemann Sums
• Apply the Fundamental Theorem of Calculus to areas and definite integrals

Text coverage:     Sections 6.2 - 6.5 of S. Tan.

Online resources:

• Week 9 Formula Sheet
• Week 9 Practice Problems
• Maple Worksheets
  • Area by Chance: The Riemann integral is usually defined as the limit of a collection of approximating sums. There after, the fundamental theorem of integral calculus provides methods for evaluating integrals without computing limits of sums. As an alternative idea, this worksheet introduces a random number generator and what is usually called "Monte-Carlo Techniques" to evaluate integrals. "Area by Chance" is a good worksheet to examine early in the introduction of the integral.
  • Techniques of Integration: Substitution and Integration by Parts: It used to be that a calculus class would study the techniques of integration so well that the student could work out how to integrate functions such as x arcsin(x). The computer can be used to help with the calculus when methods such as substitution or integration-by-parts are correct choices for integration techniques.
• Java Projects
  • Riemann sums: This applet (external link) allows the user to choose various functions and values of dx.
  • Riemann sums for integrals of a single variable: This applet is designed for experimenting with Riemann Sums and with numerical integration. It can be used to check the results of a computation of a definite integral.

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Last Modified: . School of Mathematics, Georgia Institute of Technology.