HW_4_solution.pdf - Calculus 2(Math-UA-122 Fall 2018 HW 4 Section 6.6,7.1 Solutions(80 points Due Submit online(single PDF file by before beginning of

HW_4_solution.pdf - Calculus 2(Math-UA-122 Fall 2018 HW 4...

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Calculus 2 (Math-UA-122) — Fall 2018 HW 4 — Section 6.6,7.1 — Solutions (80 points) Due: Submit online (single PDF file) by 10/1/2018 before beginning of class Show all work and write clearly and neatly in order to receive full credit. If you only write the answer with no work you will not be given any credit. Improper integrals (80 points) 1. (28 points) Evaluate the following integrals, and determine whether they converge or diverge. In the instances that you are unable to evaluate the antiderivative, use the comparison test to determine whether the integrals converge or diverge. (a) (7 points) Z 2 1 x 2 - 1 dx (b) (7 points) Z 2 1 ( x 3 - 3) 1 / 4 dx (c) (7 points) Z 2 - 2 1 4 - x 2 dx (d) (7 points) Z 1 1 1 + x 7 dx Answer: (See next page) 1
2. (14 points) For what values of p do the following integrals converge? (a) (7 points) Z e 1 1 x (ln x ) p dx (b) (7 points) Z e 1 x (ln x ) p dx Answer: (See next page) 3
3. (20 points) For any real number a , define a function Γ( a ) (called the “gamma function”) using the improper integral: Γ( a ) = Z 0 x a - 1 e - x dx (Note that for most values of a it is not possible to express this integral using elementary functions. But this integral turns out to be quite useful in many areas of math, and therefore people decided to give it a name and a special notation of its own). (a) (7 points) Using the comparison test, determine for which values of a the improper integral above converges, and for which values it diverges. Answer: The integral defining the Γ-function is improper for two reasons: (1) the range of integration continues to x = , and (2) for some values of a , the integrand is discontinuous at x = 0. To check whether Γ( a ) converges or diverges, we need to check both boundaries of integration. To this end write

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