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Clicker Question 1 Use Simpson’s Rule with n = 2 to evaluate – A. (2 2 + 1) / 3 – B. 2 / – C. (2 2 + 1) / 6 – D. 0 – E. (2 2 – 1) / 3

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Improper Integrals (2/19/14) There are two types of “improper integrals”: Type 1: Definite integral taken over a ray or the whole real line, rather than over an interval of finite length. Type 2: Definite integral of a function which becomes unbounded (i.e., “blows up”) on the interval of integration. In both types, we’re trying to integrate over an unbounded region!

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Example of Type 1 What is By this we mean, what is If this limit is a finite number, then we say the integral converges. If it is not a finite number, we say the integral diverges.

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Example of Type 2 What is As before, what this means is We make the same definition of converges and diverges.

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Simplest approach In practice, we usually dispense with the “limit as b goes to” and simply use or 0. We understand that in the numerator will cause divergence, as will 1/0, whereas in the denominator gives a value of 0 to that term. For example, then

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Clicker Question 2 What is ? – A. Converges to 1 – B. Converges to ½ – C. Converges to 2 – D. Converges to 4 – E. Diverges

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Clicker Question 3 What is ? – A. Converges to 1 – B. Converges to ½ – C. Converges to 2 – D. Converges to 4 – E. Diverges

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A word to the wise.... Most students find the idea of improper integrals somewhat mysterious at first. In particular, they tend to believe that one should get convergence (i.e., some number) or else divergence (e.g., infinity) no matter what the function is. But, in fact, IT DEPENDS ON THE FUNCTION! Two simple examples to always keep in mind are: but

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Assignment for Friday Read Section 7.8 and do Exercises 1, 3, 5, 9, 11, 15, 21 and 27 on page 527.

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