In this exercise, we use double integration to evaluate the following improper integral for a > 0 a positive constant:

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I(a) = (c) Show that I (a): So exe-ax exy dydx. -*- - e X (a) Use L'Hpital's Rule to show that f(x) = made continuous at x = 0 by assigning the value f(0) = a - 1. (b) Prove that f(x)| ex + ex for x> 1 (use the Triangle Inequality), and apply the Comparison Theorem to show that I(a) converges. noo 5 5 (d) Prove, by interchanging the order of integration, that dx -, though not defined at x = 0, can be defined and 1(a) In a lim = - T-00 ST dy