Lecture Notes: Wednesday, November 11, 1998

Tenth Homework Assignment - for discussion at recitations on Tuesday November 17, Thursday November 19, and Tuesday November 24. Because of the Thanksgiving holiday, students who normally attend recitation on Thursdays are welcome to choose a recitation to attend on Tuesday, November 24. This recitation will be devoted to Chapter 6. Our Exam #5 (Tuesday, December 1) will cover the course through Chapter 6.

Chapter 6 Section 4 (Fundamental Theorem of Calculus)

We began with a review of the definition of the definite integral in terms of Riemann sums.

Example: f(x) = 1+x^2 over [1,3] with n = 10. Write down the Riemann sums using left endpoints, right endpoints, and midpoints with a partition of [1,3] into 10 equal intervals. Look at this example in MAPLE.

 

Statement of the Fundamental Theorem of Calculus.

If f continuous on [a,b]. Then the definite integral of f(x) from a to b equals F(b) - F(a) where F is an antiderivative of f.

So the definite integral, which is a number obtained from a complicated limiting process using Riemann sums, can be calculated by finding an anti-derivative. This makes the evaluation of a definite integral quite easy in those cases in which one can write down an antiderivative of f in terms of known basic functions. (This is not always possible even if the antiderivative exists -- in such situations one has to use the Riemann sums to get numerical approximations.)

We gave the idea of the proof in terms of the instantaneous rate of change of area. See the text, page 380.

Section 6.4 # 6, 10, 16, 24, 30, 38, 42.

 

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