100%(3)3 out of 3 people found this document helpful
This preview shows page 1 - 7 out of 9 pages.
Calculus 2 (Math-UA-122) — Fall 2018HW 4 — Section 6.6,7.1 — Solutions (80 points)Due: Submit online (single PDF file) by 10/1/2018 before beginning of classShow all work and write clearly and neatly in order to receive full credit. If you only write the answer withno 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 theinstances that you are unable to evaluate the antiderivative, use the comparison test to determinewhether the integrals converge or diverge.(a)(7 points)Z∞21x2-1dx(b) (7 points)Z∞21(x3-3)1/4dx(c) (7 points)Z2-21√4-x2dx(d) (7 points)Z∞11√1 +x7dxAnswer:(See next page)1
2. (14 points) For what values ofpdo the following integrals converge?(a)(7 points)Ze11x(lnx)pdx(b) (7 points)Z∞e1x(lnx)pdxAnswer:(See next page)3
3. (20 points) For any real numbera, define a function Γ(a) (called the “gamma function”) using theimproper integral:Γ(a) =Z∞0xa-1e-xdx(Note that for most values ofait 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 togive it a name and a special notation of its own).(a) (7 points) Using the comparison test, determine for which values ofathe improper integral aboveconverges, and for which values it diverges.Answer:The integral defining the Γ-function is improper for two reasons: (1) the range ofintegration continues tox=∞, and (2) for some values ofa, the integrand is discontinuousatx= 0.To check whether Γ(a) converges or diverges, we need to check both boundaries ofintegration. To this end write