{"id":1217,"date":"2025-10-15T15:27:35","date_gmt":"2025-10-15T15:27:35","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus3\/?post_type=chapter&p=1217"},"modified":"2025-10-29T16:07:55","modified_gmt":"2025-10-29T16:07:55","slug":"applications-of-double-integrals-learn-it-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus3\/chapter\/applications-of-double-integrals-learn-it-3\/","title":{"raw":"Applications of Double Integrals: Learn It 3","rendered":"Applications of Double Integrals: Learn It 3"},"content":{"raw":"

Improper Double Integrals<\/h2>\r\n

In single-variable calculus, an improper integral 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. We briefly recall both types of improper integrals below.<\/p>\r\n\r\n

Recall: Improper Integrals\r\n[latex]\\\\[\/latex]\r\n<\/strong>A 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
    \r\n \t
  • [latex] \\displaystyle \\int_a^\\infty f(x)dx = \\displaystyle \\lim_{t\\rightarrow\\infty} \\int_a^t f(x)dx [\/latex]<\/li>\r\n \t
  • [latex] \\displaystyle \\int_{-\\infty}^b f(x)dx = \\displaystyle \\lim_{t\\rightarrow-\\infty} \\int_t^b f(x)dx [\/latex]<\/li>\r\n \t
  • [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<\/ul>\r\nIf the limits above exist, the improper integrals are said to be convergent<\/strong>.\u00a0 Otherwise, the integrals are divergent.<\/strong>\r\n\r\nA 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
      \r\n \t
    • [latex] \\displaystyle \\int_a^b f(x)dx = \\displaystyle \\lim_{t\\rightarrow b^-} \\int_a^t f(x)dx [\/latex]<\/li>\r\n \t
    • [latex] \\displaystyle \\int_a^b f(x)dx = \\displaystyle \\lim_{t\\rightarrow a^+} \\int_t^b f(x)dx [\/latex]<\/li>\r\n<\/ul>\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<\/section>An improper double integral<\/strong> is an integral [latex]\\underset{D}{\\displaystyle\\iint}fdA[\/latex] where either [latex]D[\/latex] is an unbounded region or [latex]f[\/latex] is an unbounded function. For example, [latex]D = {(x,y)\\mid |x-y| \\geq 2 }[\/latex] is an unbounded region, and the function [latex]f(x,y) = \\frac{1}{1-x^2-2y^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:\r\n
        \r\n \t
      • [latex]\\underset{D}{\\displaystyle\\iint}{xy\\,dA}[\/latex] where\u00a0[latex]D = \\{(x,y)| \\mid{x}-y\\mid\\geq2\\}[\/latex];<\/li>\r\n \t
      • [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<\/ul>\r\n

        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

        \r\n

        Fubini's theorem for improper integrals<\/h3>\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

        [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<\/section>

        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].<\/section>
        \r\n

        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

        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\"]\"The 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

        The other way to express the same region [latex]D[\/latex] is<\/p>\r\n

        [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

        [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

        Therefore, we have<\/p>\r\n

        [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<\/section>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].\r\n\r\n

        \r\n

        improper integrals on an unbounded region<\/h3>\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

        [latex]\\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<\/section>The following example shows how this theorem can be used in certain cases of improper integrals.\r\n\r\n

        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.[reveal-answer q=\"936700125\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"936700125\"]\r\n

        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}&=\\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&=\\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

        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<\/section>\r\n

        Applications to Probability<\/h3>\r\nIn 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\r\n
        \r\n

        joint density function<\/h3>\r\n

        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\u00a0joint density function<\/strong> [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

        [latex]{P}{((X,Y)\\in{D})} =\\underset{D}{\\displaystyle\\iint}{f(x,y)}{dA}[\/latex].<\/p>\r\n

        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

        [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<\/section>

        \r\n

        independent random variables<\/h3>\r\n

        The variables [latex]X[\/latex] and [latex]Y[\/latex] are said to be\u00a0independent random variables<\/span><\/strong>\u00a0if their joint density function is the product of their individual density functions:<\/p>\r\n

        [latex]{f(x,y)} = {f_1}{(x)}{f_2}{(y)}[\/latex].<\/p>\r\n\r\n<\/section>

        At Sydney\u2019s Restaurant, customers must wait an average of [latex]15[\/latex] minutes for a table. From the time they are seated until they have finished their meal requires an additional [latex]40[\/latex] 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?[reveal-answer q=\"792457310\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"792457310\"]\r\n

        Waiting times are mathematically modeled by exponential density functions, with [latex]m[\/latex] being the average waiting time, as<\/p>\r\n

        [latex]{f(t)}\\left\\{\\begin{matrix}\r\n0 & \\text{if } t < 0,\\\\\r\n{\\frac{1}{m}}{e^{-t\/m}} & \\text{if } t \\geq 0.\r\n\\end{matrix}\\right.[\/latex]<\/p>\r\n

        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

        [latex]{{f_1}(x)} = \\left\\{\\begin{matrix}\r\n0 & \\text{if } x < 0,\\\\\r\n{\\frac{1}{15}}{e^{-x\/15}} & \\text{if }x \\geq 0.\r\n\\end{matrix}\\right. \\text{and} \\\r\n{{f_2}(y)} = \\left\\{\\begin{matrix}\r\n0 & \\text{if } y < 0,\\\\\r\n{\\frac{1}{40}}{e^{-y\/40}} & \\text{if }y \\geq 0.\r\n\\end{matrix}\\right.[\/latex]<\/p>\r\n

        Clearly, the events are independent and hence the joint density function is the product of the individual functions<\/p>\r\n

        [latex]{f(x,y)} = {f_1}{(x)}{f_2}{(y)} = \\left\\{\\begin{matrix}\r\n0 & \\ \\ \\ \\ \\ \\ \\ \\ \\ \\text{if } x < 0 \\ \\text{or} \\ y < 0, \\\\\r\n{\\frac{1}{600}}{e^{-x\/15}}{e^{-y\/60}} & \\text{if } x,y \\geq 0.\r\n\\end{matrix}\\right.[\/latex]<\/p>\r\n

        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\"]\"The Figure 2.\u00a0The region of integration for a joint probability density function.<\/span>[\/caption]\r\n

        Hence, the probability that [latex](X, Y)[\/latex] is in the region [latex]D[\/latex] is<\/p>\r\n

        [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

        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} &= {\\left \\{{(x,y)}{\\mid}{0} \\ {\\leq} \\ {x} \\ {\\leq} \\ {90,0} \\ {\\leq} \\ {y} \\ {\\leq} \\ {90-x} \\right \\}}, \\\\\r\nP(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 \\\\\r\n&=\\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

        Thus, there is an [latex]83.2 \\%[\/latex] chance that a customer spends less than an hour and a half at the restaurant.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section>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.\r\n\r\n

        \r\n

        expected values<\/h3>\r\n

        In probability theory, we denote the\u00a0expected 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

        [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

        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<\/section>

        Find the expected time for the events \u2018waiting for a table\u2019 and \u2018completing the meal\u2019 in\u00a0Example \"Application to Probability\".[reveal-answer q=\"022357863\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"022357863\"]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:\r\n
        [latex]\r\n\\begin{align}\r\nE(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 \\\\\r\n&=\\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&=\\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&=\\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&=\\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&=\\frac1{600}(225)(40) \\\\\r\n&=15.\r\n\\end{align}\r\n[\/latex]<\/center>\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<\/section>","rendered":"

        Improper Double Integrals<\/h2>\n

        In single-variable calculus, an improper integral 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. We briefly recall both types of improper integrals below.<\/p>\n

        Recall: Improper Integrals
        \n[latex]\\\\[\/latex]
        \n<\/strong>A 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
          \n
        • [latex]\\displaystyle \\int_a^\\infty f(x)dx = \\displaystyle \\lim_{t\\rightarrow\\infty} \\int_a^t f(x)dx[\/latex]<\/li>\n
        • [latex]\\displaystyle \\int_{-\\infty}^b f(x)dx = \\displaystyle \\lim_{t\\rightarrow-\\infty} \\int_t^b f(x)dx[\/latex]<\/li>\n
        • [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<\/ul>\n

          If the limits above exist, the improper integrals are said to be convergent<\/strong>.\u00a0 Otherwise, the integrals are divergent.<\/strong><\/p>\n

          A 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

            \n
          • [latex]\\displaystyle \\int_a^b f(x)dx = \\displaystyle \\lim_{t\\rightarrow b^-} \\int_a^t f(x)dx[\/latex]<\/li>\n
          • [latex]\\displaystyle \\int_a^b f(x)dx = \\displaystyle \\lim_{t\\rightarrow a^+} \\int_t^b f(x)dx[\/latex]<\/li>\n<\/ul>\n

            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

            [latex]\\displaystyle \\int_a^b f(x)dx = \\displaystyle \\int_a^c f(x)dx + \\int_c^b f(x)dx[\/latex]<\/p>\n<\/section>\n

            An improper double integral<\/strong> is an integral [latex]\\underset{D}{\\displaystyle\\iint}fdA[\/latex] where either [latex]D[\/latex] is an unbounded region or [latex]f[\/latex] is an unbounded function. For example, [latex]D = {(x,y)\\mid |x-y| \\geq 2 }[\/latex] is an unbounded region, and the function [latex]f(x,y) = \\frac{1}{1-x^2-2y^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

              \n
            • [latex]\\underset{D}{\\displaystyle\\iint}{xy\\,dA}[\/latex] where\u00a0[latex]D = \\{(x,y)| \\mid{x}-y\\mid\\geq2\\}[\/latex];<\/li>\n
            • [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<\/ul>\n

              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

              \n

              Fubini’s theorem for improper integrals<\/h3>\n

              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

              [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<\/section>\n

              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].<\/section>\n
              \n

              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

              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