{"id":1064,"date":"2021-11-01T19:15:03","date_gmt":"2021-11-01T19:15:03","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=1064"},"modified":"2022-11-01T04:19:46","modified_gmt":"2022-11-01T04:19:46","slug":"improper-double-integrals","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/improper-double-integrals\/","title":{"raw":"Improper Double Integrals","rendered":"Improper Double Integrals"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul class=\"os-abstract\">\r\n \t<li><span class=\"os-abstract-content\">Solve problems involving double improper integrals.<\/span><\/li>\r\n<\/ul>\r\n<\/div>\r\nIn single-variable calculus, an <strong>improper integral\u00a0<\/strong>arises when attempting to integrate a function on an unbounded region, or attempting to integrate a function on an interval where that function is discontinuous.\u00a0 We briefly recall both types of improper integrals below.\r\n<div class=\"textbox examples\">\r\n<h3>Recall: Improper Integrals<\/h3>\r\nA <strong>type I<\/strong> improper integral arises when integrating a function [latex] f(x) [\/latex] on an unbounded interval, either [latex] [a,\\infty) [\/latex], [latex] [-\\infty,b) [\/latex], or [latex] (-\\infty,\\infty) [\/latex].\u00a0 Each of these integrals is defined in terms of a limit.\r\n<ol id=\"fs-id1167793609237\" type=\"i\">\r\n \t<li>[latex] \\displaystyle \\int_a^\\infty f(x)dx = \\displaystyle \\lim_{t\\rightarrow\\infty} \\int_a^t f(x)dx [\/latex]<\/li>\r\n \t<li>[latex] \\displaystyle \\int_{-\\infty}^b f(x)dx = \\displaystyle \\lim_{t\\rightarrow-\\infty} \\int_t^b f(x)dx [\/latex]<\/li>\r\n \t<li>[latex] \\displaystyle \\int_{-\\infty}^\\infty f(x)dx = \\displaystyle \\int_{-\\infty}^c f(x)dx + \\displaystyle \\int_c^\\infty f(x)dx [\/latex]\u00a0\u00a0 (where [latex] c [\/latex] is a real number)<\/li>\r\n<\/ol>\r\nIf the limits above exist, the improper integrals are said to be <strong>convergent<\/strong>.\u00a0 Otherwise, the integrals are <strong>divergent.<\/strong>\r\n\r\nA <strong>type II <\/strong>improper integral arises when integrating a function [latex] f(x) [\/latex] on a half-open interval [latex] [a, b) [\/latex] or [latex] (a,b] [\/latex]. In these respective cases, the improper integrals are defined as follows:\r\n<ol id=\"fs-id1167793609237\" type=\"i\">\r\n \t<li>[latex] \\displaystyle \\int_a^b f(x)dx = \\displaystyle \\lim_{t\\rightarrow b^-} \\int_a^t f(x)dx [\/latex]<\/li>\r\n \t<li>[latex] \\displaystyle \\int_a^b f(x)dx = \\displaystyle \\lim_{t\\rightarrow a^+} \\int_t^b f(x)dx [\/latex]<\/li>\r\n<\/ol>\r\nThe two definitions above can be used to define an improper integral on a closed interval [latex] [a,b] [\/latex] where there exists a [latex] c \\in [a,b] [\/latex] such that [latex] f(x) [\/latex] is discontinuous at [latex] c[\/latex].\r\n\r\n[latex] \\displaystyle \\int_a^b f(x)dx = \\displaystyle \\int_a^c f(x)dx + \\int_c^b f(x)dx [\/latex]\r\n\r\n<\/div>\r\n<p id=\"fs-id1167794067563\">An\u00a0<strong><span id=\"f783ca08-ca2f-4ebb-a559-d1b30c2f9b8c_term211\" data-type=\"term\">improper double integral<\/span><\/strong>\u00a0is an integral [latex]\\underset{D}{\\displaystyle\\iint}{f \\ dA}[\/latex] where either [latex]D[\/latex] is an unbounded region or [latex]f[\/latex] is an unbounded function. For example, [latex]{D} = {\\left \\{{(x,y)}{\\mid} \\ {\\mid}{x-y}{\\mid} \\ {\\geq} \\ {2} \\right \\}}[\/latex] is an unbounded region, and the function [latex]{f(x,y)} = {1}{\/}{(1-{x^2}-2{y^2})}[\/latex] over the ellipse [latex]{x^2} + {3y^2} \\leq {1}[\/latex] is an unbounded function. Hence, both of the following integrals are improper integrals:<\/p>\r\n\r\n<ol id=\"fs-id1167793609237\" type=\"i\">\r\n \t<li>[latex]\\underset{D}{\\displaystyle\\iint}{xy\\,dA}[\/latex] where\u00a0[latex]D = \\{(x,y)| \\mid{x}-y\\mid\\geq2\\}[\/latex];<\/li>\r\n \t<li>[latex]\\underset{D}{\\displaystyle\\iint}{\\dfrac{1}{1-x^2-2y^2}}{dA}[\/latex] where\u00a0[latex]\\large{D} = {\\left \\{{(x,y)}{\\mid}{x^2} + {3y^2} \\ {\\leq} \\ {1} \\right \\}}[\/latex].<\/li>\r\n<\/ol>\r\n<p id=\"fs-id1167793449994\">In this section we would like to deal with improper integrals of functions over rectangles or simple regions such that [latex]f[\/latex] has only finitely many discontinuities. Not all such improper integrals can be evaluated; however, a form of Fubini\u2019s theorem does apply for some types of improper integrals.<\/p>\r\n\r\n<div class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\">theorem: fubini's theorem for improper integrals<\/h3>\r\n\r\n<hr \/>\r\n\r\nIf [latex]D[\/latex] is a bounded rectangle or simple region in the plane defined by [latex]{\\left \\{{(x,y)}{:} \\ {a} \\ {\\leq} \\ {x} \\ {\\leq} \\ {b,g(x)} \\ {\\leq} \\ {y} \\ {\\leq} \\ {h(x)} \\right \\}}[\/latex] and also by [latex]{\\left \\{{(x,y)}{:} \\ {c} \\ {\\leq} \\ {y} \\ {\\leq} \\ {d,j(y)} \\ {\\leq} \\ {x} \\ {\\leq} \\ {k(y)} \\right \\}}[\/latex] and [latex]f[\/latex] is a nonnegative function on [latex]D[\/latex] with finitely many discontinuities in the interior of [latex]D[\/latex], then\r\n<p style=\"text-align: center;\">[latex]\\large{\\underset{D}{\\displaystyle\\iint}f\\,dA=\\displaystyle\\int_{x=a}^{x=b}\\displaystyle\\int_{y=g(x)}^{y=h(x)}f(x,y)dydx=\\displaystyle\\int_{y=c}^{y=d}\\displaystyle\\int_{x=j(y)}^{x=k(y)}f(x,y)dxdy.}[\/latex]<\/p>\r\n\r\n<\/div>\r\nIt is very important to note that we required that the function be nonnegative on [latex]D[\/latex] for the theorem to work. We consider only the case where the function has finitely many discontinuities inside [latex]D[\/latex].\r\n<div class=\"textbox exercises\">\r\n<h3>Example: evaluating a double improper integral<\/h3>\r\n<p id=\"fs-id1167793290993\">Consider the function [latex]{f(x,y)}={\\frac{e^y}{y}}[\/latex] over the region [latex]{D} = {\\left \\{{(x,y)}{:} \\ {0} \\ {\\leq} \\ {x} \\ {\\leq} \\ {1,x} \\ {\\leq} \\ {y} \\ {\\leq} \\ {\\sqrt{x}} \\right \\}}[\/latex].<\/p>\r\n<p id=\"fs-id1167793473203\">Notice that the function is nonnegative and continuous at all points on [latex]D[\/latex] except [latex](0, 0)[\/latex]. Use Fubini\u2019s theorem to evaluate the improper integral.<\/p>\r\n[reveal-answer q=\"854473210\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"854473210\"]\r\n\r\nFirst we plot the region [latex]D[\/latex] (Figure 1); then we express it in another way.\r\n\r\n[caption id=\"attachment_1354\" align=\"aligncenter\" width=\"454\"]<img class=\"size-full wp-image-1354\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/11\/25061144\/5-2-15.jpeg\" alt=\"The line y = x is shown, as is y = the square root of x.\" width=\"454\" height=\"460\" \/> Figure 1. The function\u00a0[latex]f[\/latex] is continuous at all points of the region\u00a0[latex]D[\/latex] except\u00a0[latex](0,0)[\/latex].[\/caption]\r\n<p id=\"fs-id1167793638851\">The other way to express the same region [latex]D[\/latex] is<\/p>\r\n<p style=\"text-align: center;\">[latex]{D} = {\\left \\{{(x,y)}{:} \\ {0} \\ {\\leq} \\ {y} \\ {\\leq} \\ {1,y^2} \\ {\\leq} \\ {x} \\ {\\leq} \\ {y} \\right \\}}[\/latex]<\/p>\r\nThus we can use Fubini\u2019s theorem for improper integrals and evaluate the integral as\r\n<p style=\"text-align: center;\">[latex]\\large{\\displaystyle\\int_{y=0}^{y=1}\\displaystyle\\int_{x=y^2}^{x=y}\\frac{e^y}y{dx}dy}.[\/latex]<\/p>\r\n<p id=\"fs-id1167793447635\">Therefore, we have<\/p>\r\n<p style=\"text-align: center;\">[latex]\\displaystyle\\int_{y=0}^{y=1}\\displaystyle\\int_{x=y^2}^{x=y}\\frac{e^y}y{dx}dy=\\displaystyle\\int_{y=0}^{y=1}\\frac{e^y}y{x}\\bigg|_{x=y^2}^{x=y}dy=\\displaystyle\\int_{y=0}^{y=1}\\frac{ey}y(y-y^2)dy=\\displaystyle\\int_0^1(ey-ye^y)dy=e-2.[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nAs mentioned before, we also have an improper integral if the region of integration is unbounded. Suppose now that the function [latex]f[\/latex] is continuous in an unbounded rectangle [latex]R[\/latex].\r\n<div class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\">theorem: improper integrals on an unbounded region<\/h3>\r\n\r\n<hr \/>\r\n\r\nIf [latex]R[\/latex] is an unbounded rectangle such as [latex]{R} = {\\left \\{{(x,y)}{:} \\ {a} \\ {\\leq} \\ {x} \\ {\\leq} \\ {\\infty, c} \\ {\\leq} \\ {y} \\ {\\leq} \\ {c} \\ {\\leq} \\ {\\infty} \\right \\}}[\/latex], then when the limit exists, we have\r\n<p style=\"text-align: center;\">[latex]\\large{\\underset{R}{\\displaystyle\\iint}f(x,y)dA=\\displaystyle\\lim_{(b,d)\\to(\\infty,\\infty)}\\displaystyle\\int_a^b\\left(\\displaystyle\\int_c^d{f}(x,y)dy\\right)dx=\\displaystyle\\lim_{(b,d)\\to(\\infty,\\infty)}\\displaystyle\\int_c^d\\left(\\displaystyle\\int_a^b{f}(x,y)dx\\right)dy}[\/latex]<\/p>\r\n\r\n<\/div>\r\nThe following example shows how this theorem can be used in certain cases of improper integrals.\r\n<div class=\"textbox exercises\">\r\n<h3>Example: evaluating a double improper integral<\/h3>\r\nEvaluate the integral [latex]\\underset{R}{\\displaystyle\\iint}{xy}{e^{{-x^2}{-y^2}}}{dA}[\/latex] where [latex]R[\/latex] is the first quadrant of the plane.\r\n\r\n[reveal-answer q=\"936700125\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"936700125\"]\r\n<p id=\"fs-id1167793633925\">The region [latex]R[\/latex] is the first quadrant of the plane, which is unbounded. So<\/p>\r\n[latex]\\begin{align}\r\n\r\n\\underset{R}{\\displaystyle\\iint}{xy}{e^{{-x^2}{-y^2}}}{dA}&amp;=\\displaystyle\\lim_{(b,d)\\to(\\infty,\\infty)}\\displaystyle\\int_{x=0}^{x=b}\\left(\\displaystyle\\int_{y=0}^{y=d}{xy}{e^{{-x^2}{-y^2}}}dy\\right)dx=\\displaystyle\\lim_{(b,d)\\to(\\infty,\\infty)}\\displaystyle\\int_{x=0}^{x=b}xe^{-x^2}dx\\displaystyle\\int_{y=0}^{y=d}{y}{e^{-y^2}}dy \\\\\r\n\r\n&amp;=\\displaystyle\\lim_{(b,d)\\to(\\infty,\\infty)}\\frac14\\left(1-e^{-b^2}\\right)\\left(1-e^{-d^2}\\right)=\\frac14\r\n\r\n\\end{align}[\/latex]\r\n<p id=\"fs-id1167794058771\">Thus, [latex]\\underset{R}{\\displaystyle\\iint}{xy}{e^{{-x^2}{-y^2}}}{dA}[\/latex] is convergent and the value is [latex]\\frac{1}{4}[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\nEvaluate the improper integral [latex]\\underset{D}{\\displaystyle\\iint}{\\dfrac{y}{\\sqrt{1-x^2-y^2}}}{dA}[\/latex] where\u00a0[latex]{D} = {\\left \\{{(x,y)}{x} \\ {\\geq} \\ {0,y} \\ {\\geq} \\ {0,x^2+y^2} \\ {\\leq} \\ {1} \\right \\}}[\/latex].\r\n\r\n[reveal-answer q=\"634193357\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"634193357\"]\r\n\r\n[latex]\\frac{\\pi}4[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n\r\n[caption]Watch the following video to see the worked solution to the above Try It[\/caption]\r\n\r\n<center><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=8197101&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=Dn3nDdB-HME&amp;video_target=tpm-plugin-0ht940zz-Dn3nDdB-HME\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/center><center>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+3\/Calc+3+transcripts\/CP5.15_transcript.html\">transcript for \u201cCP 5.15\u201d here (opens in new window).<\/a><\/center>In some situations in probability theory, we can gain insight into a problem when we are able to use double integrals over general regions. Before we go over an example with a double integral, we need to set a few definitions and become familiar with some important properties.\r\n<div class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\">definition<\/h3>\r\n\r\n<hr \/>\r\n<p id=\"fs-id1167793246025\">Consider a pair of continuous random variables [latex]X[\/latex] and [latex]Y[\/latex], such as the birthdays of two people or the number of sunny and rainy days in a month. The\u00a0<span id=\"f783ca08-ca2f-4ebb-a559-d1b30c2f9b8c_term212\" class=\"no-emphasis\" data-type=\"term\">joint density function [latex]f[\/latex] of\u00a0[latex]X[\/latex] and\u00a0[latex]Y[\/latex]\u00a0<\/span>satisfies the probability that [latex](X, Y)[\/latex] lies in a certain region [latex]D[\/latex]:<\/p>\r\n<p style=\"text-align: center;\">[latex]{P}{((X,Y)\\in{D})} =\\underset{D}{\\displaystyle\\iint}{f(x,y)}{dA}[\/latex].<\/p>\r\n<p id=\"fs-id1167793421992\">Since the probabilities can never be negative and must lie between [latex]0[\/latex] and [latex]1[\/latex], the joint density function satisfies the following inequality and equation:<\/p>\r\n<p style=\"text-align: center;\">[latex]{f(x,y)} \\ {\\geq} \\ {0}[\/latex] and [latex]\\underset{R^2}{\\displaystyle\\iint}{f(x,y)}{dA} = {1}[\/latex].<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\">definition<\/h3>\r\n\r\n<hr \/>\r\n<p id=\"fs-id1167793246025\">The variables [latex]X[\/latex] and [latex]Y[\/latex] are said to be\u00a0<span id=\"f783ca08-ca2f-4ebb-a559-d1b30c2f9b8c_term213\" class=\"no-emphasis\" data-type=\"term\">independent random variables<\/span>\u00a0if their joint density function is the product of their individual density functions:<\/p>\r\n<p style=\"text-align: center;\">[latex]{f(x,y)} = {f_1}{(x)}{f_2}{(y)}[\/latex].<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: application to probability<\/h3>\r\nAt Sydney\u2019s Restaurant, customers must wait an average of 15 minutes for a table. From the time they are seated until they have finished their meal requires an additional 40 minutes, on average. What is the probability that a customer spends less than an hour and a half at the diner, assuming that waiting for a table and completing the meal are independent events?\r\n\r\n[reveal-answer q=\"792457310\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"792457310\"]\r\n<p id=\"fs-id1167793581627\">Waiting times are mathematically modeled by exponential density functions, with [latex]m[\/latex] being the average waiting time, as<\/p>\r\n<p style=\"text-align: center;\">[latex]{f(t)}\\left\\{\\begin{matrix}\r\n0 &amp; \\text{if } t &lt; 0,\\\\\r\n{\\frac{1}{m}}{e^{-t\/m}} &amp; \\text{if } t \\geq 0.\r\n\\end{matrix}\\right.[\/latex]<\/p>\r\n<p id=\"fs-id1167793590462\">If [latex]X[\/latex] and [latex]Y[\/latex] are random variables for \u2018waiting for a table\u2019 and \u2018completing the meal,\u2019 then the probability density functions are, respectively,<\/p>\r\n<p style=\"text-align: center;\">[latex]{{f_1}(x)} = \\left\\{\\begin{matrix}\r\n0 &amp; \\text{if } x &lt; 0,\\\\\r\n{\\frac{1}{15}}{e^{-x\/15}} &amp; \\text{if }x \\geq 0.\r\n\\end{matrix}\\right. \\text{and} \\\r\n{{f_2}(y)} = \\left\\{\\begin{matrix}\r\n0 &amp; \\text{if } y &lt; 0,\\\\\r\n{\\frac{1}{40}}{e^{-y\/40}} &amp; \\text{if }y \\geq 0.\r\n\\end{matrix}\\right.[\/latex]<\/p>\r\n<p id=\"fs-id1167793619906\">Clearly, the events are independent and hence the joint density function is the product of the individual functions<\/p>\r\n<p style=\"text-align: center;\">[latex]{f(x,y)} = {f_1}{(x)}{f_2}{(y)} = \\left\\{\\begin{matrix}\r\n0 &amp; \\ \\ \\ \\ \\ \\ \\ \\ \\ \\text{if } x &lt; 0 \\ \\text{or} \\ y &lt; 0, \\\\\r\n{\\frac{1}{600}}{e^{-x\/15}}{e^{-y\/60}} &amp; \\text{if } x,y \\geq 0.\r\n\\end{matrix}\\right.[\/latex]<\/p>\r\n<p id=\"fs-id1167793928564\">We want to find the probability that the combined time [latex]X+Y[\/latex] is less than 90 minutes. In terms of geometry, it means that the region [latex]D[\/latex] is in the first quadrant bounded by the line [latex]x+y=90[\/latex] (Figure 2).<\/p>\r\n\r\n\r\n[caption id=\"attachment_1355\" align=\"aligncenter\" width=\"456\"]<img class=\"size-full wp-image-1355\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/11\/25061336\/5-2-16.jpeg\" alt=\"The line x + y = 90 is shown.\" width=\"456\" height=\"460\" \/> Figure 2.\u00a0<span class=\"os-caption\">The region of integration for a joint probability density function.<\/span>[\/caption]\r\n<p id=\"fs-id1167793927486\">Hence, the probability that [latex](X, Y)[\/latex] is in the region [latex]D[\/latex] is<\/p>\r\n<p style=\"text-align: center;\">[latex]{P} = {({X+Y} \\ {\\leq } \\ 90)} = {P}{((X,Y)\\in{D})} = {\\iint\\limits_D}{f(x,y)}{dA} = {\\iint\\limits_D}{\\frac{1}{600}}{e^{-x\/15}}{e^{-y\/40}}{dA}.[\/latex]<\/p>\r\n<p id=\"fs-id1167794292451\">Since [latex]x+y=90[\/latex] is the same as [latex]y=90-x[\/latex], we have a region of Type I, so\r\n[latex]\\hspace{6cm}\\begin{align}\r\n{D} &amp;= {\\left \\{{(x,y)}{\\mid}{0} \\ {\\leq} \\ {x} \\ {\\leq} \\ {90,0} \\ {\\leq} \\ {y} \\ {\\leq} \\ {90-x} \\right \\}}, \\\\\r\nP(X + Y\\leq90)&amp;=\\frac1{600}\\displaystyle\\int_{x=0}^{x=90}\\displaystyle\\int_{y=0}^{y=90-x}e^{-x\/15}e^{-y\/40}dxdy \\\\\r\n&amp;=\\frac1{600}\\displaystyle\\int_{x=0}^{x=90}\\displaystyle\\int_{y=0}^{y=90-x}e^{-(x\/15+y\/40)}dxdy=0.8328.\r\n\\end{align}[\/latex]<\/p>\r\n<p id=\"fs-id1167793641976\">Thus, there is an 83.2% chance that a customer spends less than an hour and a half at the restaurant.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nAnother important application in probability that can involve improper double integrals is the calculation of expected values. First we define this concept and then show an example of a calculation.\r\n<div class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\">definition<\/h3>\r\n\r\n<hr \/>\r\n<p id=\"fs-id1167793377827\">In probability theory, we denote the\u00a0<span id=\"f783ca08-ca2f-4ebb-a559-d1b30c2f9b8c_term214\" class=\"no-emphasis\" data-type=\"term\">expected values [latex]E(X)[\/latex] and\u00a0[latex]E(Y)[\/latex],\u00a0<\/span>respectively, as the most likely outcomes of the events. The expected values [latex]E(X)[\/latex] and [latex]E(Y)[\/latex] are given by<\/p>\r\n<p style=\"text-align: center;\">[latex]{E(X)} = \\underset{S}{\\displaystyle\\iint}{xf}{(x,y)}{dA}[\/latex] and [latex]{E(Y)} = \\underset{S}{\\displaystyle\\iint}{yf}{(x,y)}{dA}[\/latex],<\/p>\r\n<p id=\"fs-id1167793925058\">where [latex]S[\/latex] is the sample space of the random variables [latex]X[\/latex] and\u00a0[latex]Y[\/latex].<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: finding expected value<\/h3>\r\nFind the expected time for the events \u2018waiting for a table\u2019 and \u2018completing the meal\u2019 in\u00a0Example \"Application to Probability\".\r\n\r\n[reveal-answer q=\"022357863\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"022357863\"]\r\n\r\nUsing the first quadrant of the rectangular coordinate plane as the sample space, we have improper integrals for [latex]E(X)[\/latex] and [latex]E(Y)[\/latex]. The expected time for a table is\r\n\r\n[latex]\\begin{align}\r\n\r\nE(X)&amp;=\\underset{S}{\\displaystyle\\iint}x\\frac1{600}e^{-x\/15}e^{-y\/40}dA=\\frac1{600}\\displaystyle\\int_{x=0}^{x=\\infty}\\displaystyle\\int_{y=0}^{y=\\infty}xe^{-x\/15}e^{-y\/40}dA \\\\\r\n\r\n&amp;=\\frac1{600}\\displaystyle\\lim_{(a,b)\\to(\\infty,\\infty)}\\displaystyle\\int_{x=0}^{x=a}\\displaystyle\\int_{y=0}^{y=b}xe^{-x\/15}e^{-y\/40}dxdy \\\\\r\n\r\n&amp;=\\frac1{600}\\left(\\displaystyle\\lim_{a\\to\\infty}\\displaystyle\\int_{x=0}^{x=a}xe^{-x\/15}dx\\right)\\left(\\displaystyle\\int_{y=0}^{y=b}e^{-y\/40}dy\\right) \\\\\r\n\r\n&amp;=\\frac1{600}\\left((\\displaystyle\\lim_{a\\to\\infty}(-15e^{-x\/15}(x+15)))\\Bigg|_{x=0}^{x=a}\\right)\\left((\\displaystyle\\lim_{b\\to\\infty}(-40e^{-y\/40}))\\Bigg|_{y=0}^{y=b}\\right) \\\\\r\n\r\n&amp;=\\frac1{600}\\left((\\displaystyle\\lim_{a\\to\\infty}(-15e^{-a\/15}(x+15)+225)\\right)\\left((\\displaystyle\\lim_{b\\to\\infty}(-40e^{-b\/40}+40)\\right) \\\\\r\n\r\n&amp;=\\frac1{600}(225)(40) \\\\\r\n\r\n&amp;=15.\r\n\r\n\\end{align}[\/latex]\r\n\r\nA similar calculation shows that [latex]E(Y)=40[\/latex]. This means that the expected values of the two random events are the average waiting time and the average dining time, respectively.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n<p id=\"fs-id1167793499076\">The joint density function for two random variables [latex]X[\/latex] and [latex]Y[\/latex] is given by<\/p>\r\n<p style=\"text-align: left;\">[latex]\\hspace{6cm}f(x,y) = \\begin{align}\r\n&amp;\\frac1{16,250}(x^2+y^2) &amp;\\,&amp;\\text{ if } 0 \\leq x \\leq 15,0 \\leq y \\leq 10 \\\\\r\n&amp;0 &amp;\\,&amp;\\text{otherwise}\r\n\\end{align}[\/latex]<\/p>\r\n<p id=\"fs-id1167793462449\">Find the probability that [latex]X[\/latex] is at most 10 and [latex]Y[\/latex] is at least 5.<\/p>\r\n[reveal-answer q=\"889466210\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"889466210\"]\r\n\r\n[latex]\\frac{55}{72}\\approx0.7638[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<ul class=\"os-abstract\">\n<li><span class=\"os-abstract-content\">Solve problems involving double improper integrals.<\/span><\/li>\n<\/ul>\n<\/div>\n<p>In single-variable calculus, an <strong>improper integral\u00a0<\/strong>arises when attempting to integrate a function on an unbounded region, or attempting to integrate a function on an interval where that function is discontinuous.\u00a0 We briefly recall both types of improper integrals below.<\/p>\n<div class=\"textbox examples\">\n<h3>Recall: Improper Integrals<\/h3>\n<p>A <strong>type I<\/strong> improper integral arises when integrating a function [latex]f(x)[\/latex] on an unbounded interval, either [latex][a,\\infty)[\/latex], [latex][-\\infty,b)[\/latex], or [latex](-\\infty,\\infty)[\/latex].\u00a0 Each of these integrals is defined in terms of a limit.<\/p>\n<ol id=\"fs-id1167793609237\" type=\"i\">\n<li>[latex]\\displaystyle \\int_a^\\infty f(x)dx = \\displaystyle \\lim_{t\\rightarrow\\infty} \\int_a^t f(x)dx[\/latex]<\/li>\n<li>[latex]\\displaystyle \\int_{-\\infty}^b f(x)dx = \\displaystyle \\lim_{t\\rightarrow-\\infty} \\int_t^b f(x)dx[\/latex]<\/li>\n<li>[latex]\\displaystyle \\int_{-\\infty}^\\infty f(x)dx = \\displaystyle \\int_{-\\infty}^c f(x)dx + \\displaystyle \\int_c^\\infty f(x)dx[\/latex]\u00a0\u00a0 (where [latex]c[\/latex] is a real number)<\/li>\n<\/ol>\n<p>If the limits above exist, the improper integrals are said to be <strong>convergent<\/strong>.\u00a0 Otherwise, the integrals are <strong>divergent.<\/strong><\/p>\n<p>A <strong>type II <\/strong>improper integral arises when integrating a function [latex]f(x)[\/latex] on a half-open interval [latex][a, b)[\/latex] or [latex](a,b][\/latex]. In these respective cases, the improper integrals are defined as follows:<\/p>\n<ol id=\"fs-id1167793609237\" type=\"i\">\n<li>[latex]\\displaystyle \\int_a^b f(x)dx = \\displaystyle \\lim_{t\\rightarrow b^-} \\int_a^t f(x)dx[\/latex]<\/li>\n<li>[latex]\\displaystyle \\int_a^b f(x)dx = \\displaystyle \\lim_{t\\rightarrow a^+} \\int_t^b f(x)dx[\/latex]<\/li>\n<\/ol>\n<p>The two definitions above can be used to define an improper integral on a closed interval [latex][a,b][\/latex] where there exists a [latex]c \\in [a,b][\/latex] such that [latex]f(x)[\/latex] is discontinuous at [latex]c[\/latex].<\/p>\n<p>[latex]\\displaystyle \\int_a^b f(x)dx = \\displaystyle \\int_a^c f(x)dx + \\int_c^b f(x)dx[\/latex]<\/p>\n<\/div>\n<p id=\"fs-id1167794067563\">An\u00a0<strong><span id=\"f783ca08-ca2f-4ebb-a559-d1b30c2f9b8c_term211\" data-type=\"term\">improper double integral<\/span><\/strong>\u00a0is an integral [latex]\\underset{D}{\\displaystyle\\iint}{f \\ dA}[\/latex] where either [latex]D[\/latex] is an unbounded region or [latex]f[\/latex] is an unbounded function. For example, [latex]{D} = {\\left \\{{(x,y)}{\\mid} \\ {\\mid}{x-y}{\\mid} \\ {\\geq} \\ {2} \\right \\}}[\/latex] is an unbounded region, and the function [latex]{f(x,y)} = {1}{\/}{(1-{x^2}-2{y^2})}[\/latex] over the ellipse [latex]{x^2} + {3y^2} \\leq {1}[\/latex] is an unbounded function. Hence, both of the following integrals are improper integrals:<\/p>\n<ol id=\"fs-id1167793609237\" type=\"i\">\n<li>[latex]\\underset{D}{\\displaystyle\\iint}{xy\\,dA}[\/latex] where\u00a0[latex]D = \\{(x,y)| \\mid{x}-y\\mid\\geq2\\}[\/latex];<\/li>\n<li>[latex]\\underset{D}{\\displaystyle\\iint}{\\dfrac{1}{1-x^2-2y^2}}{dA}[\/latex] where\u00a0[latex]\\large{D} = {\\left \\{{(x,y)}{\\mid}{x^2} + {3y^2} \\ {\\leq} \\ {1} \\right \\}}[\/latex].<\/li>\n<\/ol>\n<p id=\"fs-id1167793449994\">In this section we would like to deal with improper integrals of functions over rectangles or simple regions such that [latex]f[\/latex] has only finitely many discontinuities. Not all such improper integrals can be evaluated; however, a form of Fubini\u2019s theorem does apply for some types of improper integrals.<\/p>\n<div class=\"textbox shaded\">\n<h3 style=\"text-align: center;\">theorem: fubini&#8217;s theorem for improper integrals<\/h3>\n<hr \/>\n<p>If [latex]D[\/latex] is a bounded rectangle or simple region in the plane defined by [latex]{\\left \\{{(x,y)}{:} \\ {a} \\ {\\leq} \\ {x} \\ {\\leq} \\ {b,g(x)} \\ {\\leq} \\ {y} \\ {\\leq} \\ {h(x)} \\right \\}}[\/latex] and also by [latex]{\\left \\{{(x,y)}{:} \\ {c} \\ {\\leq} \\ {y} \\ {\\leq} \\ {d,j(y)} \\ {\\leq} \\ {x} \\ {\\leq} \\ {k(y)} \\right \\}}[\/latex] and [latex]f[\/latex] is a nonnegative function on [latex]D[\/latex] with finitely many discontinuities in the interior of [latex]D[\/latex], then<\/p>\n<p style=\"text-align: center;\">[latex]\\large{\\underset{D}{\\displaystyle\\iint}f\\,dA=\\displaystyle\\int_{x=a}^{x=b}\\displaystyle\\int_{y=g(x)}^{y=h(x)}f(x,y)dydx=\\displaystyle\\int_{y=c}^{y=d}\\displaystyle\\int_{x=j(y)}^{x=k(y)}f(x,y)dxdy.}[\/latex]<\/p>\n<\/div>\n<p>It is very important to note that we required that the function be nonnegative on [latex]D[\/latex] for the theorem to work. We consider only the case where the function has finitely many discontinuities inside [latex]D[\/latex].<\/p>\n<div class=\"textbox exercises\">\n<h3>Example: evaluating a double improper integral<\/h3>\n<p id=\"fs-id1167793290993\">Consider the function [latex]{f(x,y)}={\\frac{e^y}{y}}[\/latex] over the region [latex]{D} = {\\left \\{{(x,y)}{:} \\ {0} \\ {\\leq} \\ {x} \\ {\\leq} \\ {1,x} \\ {\\leq} \\ {y} \\ {\\leq} \\ {\\sqrt{x}} \\right \\}}[\/latex].<\/p>\n<p id=\"fs-id1167793473203\">Notice that the function is nonnegative and continuous at all points on [latex]D[\/latex] except [latex](0, 0)[\/latex]. Use Fubini\u2019s theorem to evaluate the improper integral.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q854473210\">Show Solution<\/span><\/p>\n<div id=\"q854473210\" class=\"hidden-answer\" style=\"display: none\">\n<p>First we plot the region [latex]D[\/latex] (Figure 1); then we express it in another way.<\/p>\n<div id=\"attachment_1354\" style=\"width: 464px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1354\" class=\"size-full wp-image-1354\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/11\/25061144\/5-2-15.jpeg\" alt=\"The line y = x is shown, as is y = the square root of x.\" width=\"454\" height=\"460\" \/><\/p>\n<p id=\"caption-attachment-1354\" class=\"wp-caption-text\">Figure 1. The function\u00a0[latex]f[\/latex] is continuous at all points of the region\u00a0[latex]D[\/latex] except\u00a0[latex](0,0)[\/latex].<\/p>\n<\/div>\n<p id=\"fs-id1167793638851\">The other way to express the same region [latex]D[\/latex] is<\/p>\n<p style=\"text-align: center;\">[latex]{D} = {\\left \\{{(x,y)}{:} \\ {0} \\ {\\leq} \\ {y} \\ {\\leq} \\ {1,y^2} \\ {\\leq} \\ {x} \\ {\\leq} \\ {y} \\right \\}}[\/latex]<\/p>\n<p>Thus we can use Fubini\u2019s theorem for improper integrals and evaluate the integral as<\/p>\n<p style=\"text-align: center;\">[latex]\\large{\\displaystyle\\int_{y=0}^{y=1}\\displaystyle\\int_{x=y^2}^{x=y}\\frac{e^y}y{dx}dy}.[\/latex]<\/p>\n<p id=\"fs-id1167793447635\">Therefore, we have<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle\\int_{y=0}^{y=1}\\displaystyle\\int_{x=y^2}^{x=y}\\frac{e^y}y{dx}dy=\\displaystyle\\int_{y=0}^{y=1}\\frac{e^y}y{x}\\bigg|_{x=y^2}^{x=y}dy=\\displaystyle\\int_{y=0}^{y=1}\\frac{ey}y(y-y^2)dy=\\displaystyle\\int_0^1(ey-ye^y)dy=e-2.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>As mentioned before, we also have an improper integral if the region of integration is unbounded. Suppose now that the function [latex]f[\/latex] is continuous in an unbounded rectangle [latex]R[\/latex].<\/p>\n<div class=\"textbox shaded\">\n<h3 style=\"text-align: center;\">theorem: improper integrals on an unbounded region<\/h3>\n<hr \/>\n<p>If [latex]R[\/latex] is an unbounded rectangle such as [latex]{R} = {\\left \\{{(x,y)}{:} \\ {a} \\ {\\leq} \\ {x} \\ {\\leq} \\ {\\infty, c} \\ {\\leq} \\ {y} \\ {\\leq} \\ {c} \\ {\\leq} \\ {\\infty} \\right \\}}[\/latex], then when the limit exists, we have<\/p>\n<p style=\"text-align: center;\">[latex]\\large{\\underset{R}{\\displaystyle\\iint}f(x,y)dA=\\displaystyle\\lim_{(b,d)\\to(\\infty,\\infty)}\\displaystyle\\int_a^b\\left(\\displaystyle\\int_c^d{f}(x,y)dy\\right)dx=\\displaystyle\\lim_{(b,d)\\to(\\infty,\\infty)}\\displaystyle\\int_c^d\\left(\\displaystyle\\int_a^b{f}(x,y)dx\\right)dy}[\/latex]<\/p>\n<\/div>\n<p>The following example shows how this theorem can be used in certain cases of improper integrals.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example: evaluating a double improper integral<\/h3>\n<p>Evaluate the integral [latex]\\underset{R}{\\displaystyle\\iint}{xy}{e^{{-x^2}{-y^2}}}{dA}[\/latex] where [latex]R[\/latex] is the first quadrant of the plane.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q936700125\">Show Solution<\/span><\/p>\n<div id=\"q936700125\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167793633925\">The region [latex]R[\/latex] is the first quadrant of the plane, which is unbounded. So<\/p>\n<p>[latex]\\begin{align}    \\underset{R}{\\displaystyle\\iint}{xy}{e^{{-x^2}{-y^2}}}{dA}&=\\displaystyle\\lim_{(b,d)\\to(\\infty,\\infty)}\\displaystyle\\int_{x=0}^{x=b}\\left(\\displaystyle\\int_{y=0}^{y=d}{xy}{e^{{-x^2}{-y^2}}}dy\\right)dx=\\displaystyle\\lim_{(b,d)\\to(\\infty,\\infty)}\\displaystyle\\int_{x=0}^{x=b}xe^{-x^2}dx\\displaystyle\\int_{y=0}^{y=d}{y}{e^{-y^2}}dy \\\\    &=\\displaystyle\\lim_{(b,d)\\to(\\infty,\\infty)}\\frac14\\left(1-e^{-b^2}\\right)\\left(1-e^{-d^2}\\right)=\\frac14    \\end{align}[\/latex]<\/p>\n<p id=\"fs-id1167794058771\">Thus, [latex]\\underset{R}{\\displaystyle\\iint}{xy}{e^{{-x^2}{-y^2}}}{dA}[\/latex] is convergent and the value is [latex]\\frac{1}{4}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p>Evaluate the improper integral [latex]\\underset{D}{\\displaystyle\\iint}{\\dfrac{y}{\\sqrt{1-x^2-y^2}}}{dA}[\/latex] where\u00a0[latex]{D} = {\\left \\{{(x,y)}{x} \\ {\\geq} \\ {0,y} \\ {\\geq} \\ {0,x^2+y^2} \\ {\\leq} \\ {1} \\right \\}}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q634193357\">Show Solution<\/span><\/p>\n<div id=\"q634193357\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\frac{\\pi}4[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Watch the following video to see the worked solution to the above Try It<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=8197101&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=Dn3nDdB-HME&amp;video_target=tpm-plugin-0ht940zz-Dn3nDdB-HME\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/div>\n<div style=\"text-align: center;\">You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+3\/Calc+3+transcripts\/CP5.15_transcript.html\">transcript for \u201cCP 5.15\u201d here (opens in new window).<\/a><\/div>\n<p>In some situations in probability theory, we can gain insight into a problem when we are able to use double integrals over general regions. Before we go over an example with a double integral, we need to set a few definitions and become familiar with some important properties.<\/p>\n<div class=\"textbox shaded\">\n<h3 style=\"text-align: center;\">definition<\/h3>\n<hr \/>\n<p id=\"fs-id1167793246025\">Consider a pair of continuous random variables [latex]X[\/latex] and [latex]Y[\/latex], such as the birthdays of two people or the number of sunny and rainy days in a month. The\u00a0<span id=\"f783ca08-ca2f-4ebb-a559-d1b30c2f9b8c_term212\" class=\"no-emphasis\" data-type=\"term\">joint density function [latex]f[\/latex] of\u00a0[latex]X[\/latex] and\u00a0[latex]Y[\/latex]\u00a0<\/span>satisfies the probability that [latex](X, Y)[\/latex] lies in a certain region [latex]D[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]{P}{((X,Y)\\in{D})} =\\underset{D}{\\displaystyle\\iint}{f(x,y)}{dA}[\/latex].<\/p>\n<p id=\"fs-id1167793421992\">Since the probabilities can never be negative and must lie between [latex]0[\/latex] and [latex]1[\/latex], the joint density function satisfies the following inequality and equation:<\/p>\n<p style=\"text-align: center;\">[latex]{f(x,y)} \\ {\\geq} \\ {0}[\/latex] and [latex]\\underset{R^2}{\\displaystyle\\iint}{f(x,y)}{dA} = {1}[\/latex].<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<h3 style=\"text-align: center;\">definition<\/h3>\n<hr \/>\n<p id=\"fs-id1167793246025\">The variables [latex]X[\/latex] and [latex]Y[\/latex] are said to be\u00a0<span id=\"f783ca08-ca2f-4ebb-a559-d1b30c2f9b8c_term213\" class=\"no-emphasis\" data-type=\"term\">independent random variables<\/span>\u00a0if their joint density function is the product of their individual density functions:<\/p>\n<p style=\"text-align: center;\">[latex]{f(x,y)} = {f_1}{(x)}{f_2}{(y)}[\/latex].<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: application to probability<\/h3>\n<p>At Sydney\u2019s Restaurant, customers must wait an average of 15 minutes for a table. From the time they are seated until they have finished their meal requires an additional 40 minutes, on average. What is the probability that a customer spends less than an hour and a half at the diner, assuming that waiting for a table and completing the meal are independent events?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q792457310\">Show Solution<\/span><\/p>\n<div id=\"q792457310\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167793581627\">Waiting times are mathematically modeled by exponential density functions, with [latex]m[\/latex] being the average waiting time, as<\/p>\n<p style=\"text-align: center;\">[latex]{f(t)}\\left\\{\\begin{matrix}  0 & \\text{if } t < 0,\\\\  {\\frac{1}{m}}{e^{-t\/m}} & \\text{if } t \\geq 0.  \\end{matrix}\\right.[\/latex]<\/p>\n<p id=\"fs-id1167793590462\">If [latex]X[\/latex] and [latex]Y[\/latex] are random variables for \u2018waiting for a table\u2019 and \u2018completing the meal,\u2019 then the probability density functions are, respectively,<\/p>\n<p style=\"text-align: center;\">[latex]{{f_1}(x)} = \\left\\{\\begin{matrix}  0 & \\text{if } x < 0,\\\\  {\\frac{1}{15}}{e^{-x\/15}} & \\text{if }x \\geq 0.  \\end{matrix}\\right. \\text{and} \\  {{f_2}(y)} = \\left\\{\\begin{matrix}  0 & \\text{if } y < 0,\\\\  {\\frac{1}{40}}{e^{-y\/40}} & \\text{if }y \\geq 0.  \\end{matrix}\\right.[\/latex]<\/p>\n<p id=\"fs-id1167793619906\">Clearly, the events are independent and hence the joint density function is the product of the individual functions<\/p>\n<p style=\"text-align: center;\">[latex]{f(x,y)} = {f_1}{(x)}{f_2}{(y)} = \\left\\{\\begin{matrix}  0 & \\ \\ \\ \\ \\ \\ \\ \\ \\ \\text{if } x < 0 \\ \\text{or} \\ y < 0, \\\\  {\\frac{1}{600}}{e^{-x\/15}}{e^{-y\/60}} & \\text{if } x,y \\geq 0.  \\end{matrix}\\right.[\/latex]<\/p>\n<p id=\"fs-id1167793928564\">We want to find the probability that the combined time [latex]X+Y[\/latex] is less than 90 minutes. In terms of geometry, it means that the region [latex]D[\/latex] is in the first quadrant bounded by the line [latex]x+y=90[\/latex] (Figure 2).<\/p>\n<div id=\"attachment_1355\" style=\"width: 466px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1355\" class=\"size-full wp-image-1355\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/11\/25061336\/5-2-16.jpeg\" alt=\"The line x + y = 90 is shown.\" width=\"456\" height=\"460\" \/><\/p>\n<p id=\"caption-attachment-1355\" class=\"wp-caption-text\">Figure 2.\u00a0<span class=\"os-caption\">The region of integration for a joint probability density function.<\/span><\/p>\n<\/div>\n<p id=\"fs-id1167793927486\">Hence, the probability that [latex](X, Y)[\/latex] is in the region [latex]D[\/latex] is<\/p>\n<p style=\"text-align: center;\">[latex]{P} = {({X+Y} \\ {\\leq } \\ 90)} = {P}{((X,Y)\\in{D})} = {\\iint\\limits_D}{f(x,y)}{dA} = {\\iint\\limits_D}{\\frac{1}{600}}{e^{-x\/15}}{e^{-y\/40}}{dA}.[\/latex]<\/p>\n<p id=\"fs-id1167794292451\">Since [latex]x+y=90[\/latex] is the same as [latex]y=90-x[\/latex], we have a region of Type I, so<br \/>\n[latex]\\hspace{6cm}\\begin{align}  {D} &= {\\left \\{{(x,y)}{\\mid}{0} \\ {\\leq} \\ {x} \\ {\\leq} \\ {90,0} \\ {\\leq} \\ {y} \\ {\\leq} \\ {90-x} \\right \\}}, \\\\  P(X + Y\\leq90)&=\\frac1{600}\\displaystyle\\int_{x=0}^{x=90}\\displaystyle\\int_{y=0}^{y=90-x}e^{-x\/15}e^{-y\/40}dxdy \\\\  &=\\frac1{600}\\displaystyle\\int_{x=0}^{x=90}\\displaystyle\\int_{y=0}^{y=90-x}e^{-(x\/15+y\/40)}dxdy=0.8328.  \\end{align}[\/latex]<\/p>\n<p id=\"fs-id1167793641976\">Thus, there is an 83.2% chance that a customer spends less than an hour and a half at the restaurant.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Another important application in probability that can involve improper double integrals is the calculation of expected values. First we define this concept and then show an example of a calculation.<\/p>\n<div class=\"textbox shaded\">\n<h3 style=\"text-align: center;\">definition<\/h3>\n<hr \/>\n<p id=\"fs-id1167793377827\">In probability theory, we denote the\u00a0<span id=\"f783ca08-ca2f-4ebb-a559-d1b30c2f9b8c_term214\" class=\"no-emphasis\" data-type=\"term\">expected values [latex]E(X)[\/latex] and\u00a0[latex]E(Y)[\/latex],\u00a0<\/span>respectively, as the most likely outcomes of the events. The expected values [latex]E(X)[\/latex] and [latex]E(Y)[\/latex] are given by<\/p>\n<p style=\"text-align: center;\">[latex]{E(X)} = \\underset{S}{\\displaystyle\\iint}{xf}{(x,y)}{dA}[\/latex] and [latex]{E(Y)} = \\underset{S}{\\displaystyle\\iint}{yf}{(x,y)}{dA}[\/latex],<\/p>\n<p id=\"fs-id1167793925058\">where [latex]S[\/latex] is the sample space of the random variables [latex]X[\/latex] and\u00a0[latex]Y[\/latex].<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: finding expected value<\/h3>\n<p>Find the expected time for the events \u2018waiting for a table\u2019 and \u2018completing the meal\u2019 in\u00a0Example &#8220;Application to Probability&#8221;.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q022357863\">Show Solution<\/span><\/p>\n<div id=\"q022357863\" class=\"hidden-answer\" style=\"display: none\">\n<p>Using the first quadrant of the rectangular coordinate plane as the sample space, we have improper integrals for [latex]E(X)[\/latex] and [latex]E(Y)[\/latex]. The expected time for a table is<\/p>\n<p>[latex]\\begin{align}    E(X)&=\\underset{S}{\\displaystyle\\iint}x\\frac1{600}e^{-x\/15}e^{-y\/40}dA=\\frac1{600}\\displaystyle\\int_{x=0}^{x=\\infty}\\displaystyle\\int_{y=0}^{y=\\infty}xe^{-x\/15}e^{-y\/40}dA \\\\    &=\\frac1{600}\\displaystyle\\lim_{(a,b)\\to(\\infty,\\infty)}\\displaystyle\\int_{x=0}^{x=a}\\displaystyle\\int_{y=0}^{y=b}xe^{-x\/15}e^{-y\/40}dxdy \\\\    &=\\frac1{600}\\left(\\displaystyle\\lim_{a\\to\\infty}\\displaystyle\\int_{x=0}^{x=a}xe^{-x\/15}dx\\right)\\left(\\displaystyle\\int_{y=0}^{y=b}e^{-y\/40}dy\\right) \\\\    &=\\frac1{600}\\left((\\displaystyle\\lim_{a\\to\\infty}(-15e^{-x\/15}(x+15)))\\Bigg|_{x=0}^{x=a}\\right)\\left((\\displaystyle\\lim_{b\\to\\infty}(-40e^{-y\/40}))\\Bigg|_{y=0}^{y=b}\\right) \\\\    &=\\frac1{600}\\left((\\displaystyle\\lim_{a\\to\\infty}(-15e^{-a\/15}(x+15)+225)\\right)\\left((\\displaystyle\\lim_{b\\to\\infty}(-40e^{-b\/40}+40)\\right) \\\\    &=\\frac1{600}(225)(40) \\\\    &=15.    \\end{align}[\/latex]<\/p>\n<p>A similar calculation shows that [latex]E(Y)=40[\/latex]. This means that the expected values of the two random events are the average waiting time and the average dining time, respectively.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p id=\"fs-id1167793499076\">The joint density function for two random variables [latex]X[\/latex] and [latex]Y[\/latex] is given by<\/p>\n<p style=\"text-align: left;\">[latex]\\hspace{6cm}f(x,y) = \\begin{align}  &\\frac1{16,250}(x^2+y^2) &\\,&\\text{ if } 0 \\leq x \\leq 15,0 \\leq y \\leq 10 \\\\  &0 &\\,&\\text{otherwise}  \\end{align}[\/latex]<\/p>\n<p id=\"fs-id1167793462449\">Find the probability that [latex]X[\/latex] is at most 10 and [latex]Y[\/latex] is at least 5.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q889466210\">Show Solution<\/span><\/p>\n<div id=\"q889466210\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\frac{55}{72}\\approx0.7638[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-1064\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>CP 5.15. <strong>Authored by<\/strong>: Ryan Melton. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Calculus Volume 3. <strong>Authored by<\/strong>: Gilbert Strang, Edwin (Jed) Herman. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction\">https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/4.0\/\">CC BY-NC-SA: Attribution-NonCommercial-ShareAlike<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":428269,"menu_order":10,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Calculus Volume 3\",\"author\":\"Gilbert Strang, Edwin (Jed) Herman\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction\",\"project\":\"\",\"license\":\"cc-by-nc-sa\",\"license_terms\":\"Access for free at 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