Lecture Notes: Monday, November 9, 1998

Ninth Homework Assignment - for discussion at recitations Tuesday November 10, and Thursday November 12.

Chapter 6 Section 3 (Area and the Definite Integral)

Two problems:

• 1. Area under the graph of a the function f(x) = x^2 and above the x-axis, between x=0 and x=10.
• 2. Rod of density x along x-axis from 0 to 10. Moment of inertia about x=0.

Approximate solutions to both problems lead to the same Riemann sum. The limit of this Riemann sum as the partition of the interval from 0 to 10 gets finer and finer is a number called the Riemann integral of f(x) from 0 to 10.

Formal definition of definite integral.

Theorem: Continuous functions are integrable.

Area interpretation if f(x) changes sign.

Approximations using

• midpoints
• left endpoints
• right endpoints

Problem 6.3 #10.

After starting this problem by hand, we used MAPLE and "with(student)."

We also used right and left endpoints and saw how to use these estimates to get bounds on the integral .

MAPLE Demonstration #11 Please work through this demonstration.

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