Homework 16 – Improper Integrals 1) Which of the following integrals is improper? Why or why not? Do not evaluate the integrals. a) 2130dxx b) 0.21dxx c) 1xedx d) 10xedx e) 20secxdx f) 0sinxdx g) 10sinxdx h) 1023dxx i) 1lnxdx j) 30lnxdx 2) Determine whether each of the following improper integrals converges and, if so, evaluate it. a) 19 201dxx b) 40.0001tedt c) 520 190dxx d) 404dxx e) 0sinxdx f) 10lnxdx g) 120lnxdxx h) 2xxedx 3) Use the comparison Test to determine whether or not each of the following integrals converges. a) 1512dxx b) 120xedxx c) 411dxxx 4) The solid region G obtained by rotating the region below the graph of 1yx about the x‐axis for 1x is called Gabriel’s Horn. Show that the volume of G is finite and that the surface area of G is infinite. 5) The Laplace Transform of a function, f(x), is the function Lf(s) of the variable s defined by the improper integral 0sxLf s f x e dx if it converges. a) Show that if fx C, where C is a constant, then CLf ss for s > 0. b) Compute Lf s, where xfx
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