{"id":32,"date":"2022-01-10T15:06:33","date_gmt":"2022-01-10T20:06:33","guid":{"rendered":"https:\/\/pressbooks.library.ryerson.ca\/multivariatecalculus\/?post_type=chapter&p=32"},"modified":"2022-03-03T22:26:33","modified_gmt":"2022-03-04T03:26:33","slug":"chapter-2","status":"publish","type":"chapter","link":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/chapter\/chapter-2\/","title":{"raw":"Unit 5: Double Integral Over Rectangular Regions","rendered":"Unit 5: Double Integral Over Rectangular Regions"},"content":{"raw":"

The Concept<\/h2>\r\nThe definition of the single integral<\/strong> in 2D space is as follows: given a single-variable function<\/strong> [latex]y=f(x)[\/latex] that is continuous on the interval [latex][a,b][\/latex], we divide the interval into [latex]n[\/latex] subintervals of equal width, [latex]x[\/latex], and from each interval choose a point, [latex]x_i[\/latex]. Definite integral,<\/strong> [latex]\\int_a^bf(x)dx[\/latex], represents the area inbetween the curve, [latex]y=f(x)[\/latex], and [latex]x[\/latex]-axis. Riemann sum<\/strong> helps to approximate such areas, that is,\r\n

[latex]\\int_a^bf(x)dx \\approx \\sum_{i=1}^n f(x_i) \\Delta x[\/latex],\r\nwhere [latex]\\Delta x = \\frac{b-a}{n}[\/latex] and [latex]x_i=a + i\\Delta x[\/latex]. The larger [latex]n[\/latex] is, the better the estimation is. Thus, the limit of the Riemann sum defines the definite integral,<\/p>\r\n

[latex]\\int_a^bf(x)dx =\\lim_{n \\to \\infty} \\sum_{i=1}^n f(x_i) \\Delta x = \\sum_{i=1}^{\\infty} f(x_i)\\,\\Delta x[\/latex].<\/p>\r\n \r\n\r\nSimilar to the single integral, the double integral<\/strong> in 3D, [latex]\\iint_R f(x,y) dA[\/latex], is equal to the volume under the surface of the two-variable function<\/strong> [latex]z = f(x,y)[\/latex] and above the region [latex]R[\/latex] on the [latex]xy[\/latex]-plane. Here, we consider this region has a very simple shape, rectangle, and use [latex]R[\/latex] to denote it. The [latex]x[\/latex] coordinate of this rectangle changes from [latex]a[\/latex] to [latex]b[\/latex], and [latex]y[\/latex] coordinate changes from [latex]c[\/latex] to [latex]d[\/latex], denoted as [latex]R=[a, b]\\times[c,d][\/latex]. As in the case of the single integral, a double integral<\/strong> is defined as the limit of a Reimann sum<\/strong>, i.e.,\r\n

[latex]\\iint_R f(x,y) dA=\\lim_{m,n \\to \\infty}\\sum_{i=1}^{m} \\sum_{j=1}^{n} f(x_{i},y_{j}) \\Delta A[\/latex]<\/p>\r\nwhere [latex]\\Delta A=\\Delta x \\Delta y, \\Delta x=\\frac{b-a}{n}, \\Delta y=\\frac{d-c}{m}, x_i=a + i\\Delta x[\/latex] and [latex]y_i=c + i\\Delta y[\/latex].\r\n

The Plot<\/h2>\r\nNow, you should engage with the 3D plot below to understand double integrals over rectangular regions[footnote]Made with GeoGebra, licensed Creative Commons CC BY-NC-SA 4.0.[\/footnote]. Follow the steps below to apply changes to the plot and observe the effects:\r\n
    \r\n \t
  1. Input a function of two variables into the [latex]f(x,y)[\/latex] input function section.<\/li>\r\n \t
  2. Move the [latex]n[\/latex]-slide around to decide the subregions of the rectangular region, [latex]R[\/latex], and we consider the subregions are squares.<\/li>\r\n \t
  3. Pick xmin, xmax, ymin, and ymax points for your domain\/bounds of the rectangular region, [latex]R[\/latex].<\/li>\r\n \t
  4. Use the [latex]k[\/latex]-slider to choose which square-shaped subregion you\u2019d like to highlight.<\/li>\r\n \t
  5. Use the checkboxes to show either all of the rectangular prisms compared to just the one you are highlighting, as well as whether to see the graph or not.<\/li>\r\n \t
  6. By changing your view and hovering over the plot, you can see a 2D representation of the rectangular area. Additionally, the double integral is dynamically calculated at the bottom.<\/li>\r\n<\/ol>\r\n[h5p id=\"16\"]<\/span>\r\n\r\n
    <\/header>\r\n
    \r\n
    \r\n

    Self-Checking Questions<\/h2>\r\n<\/header>\r\n
    \r\n\r\nCheck your understanding by solving the following questions[footnote]Gilbert Strang, Edwin \u201cJed\u201d Herman,\u00a0 OpenStax,\u00a0Calculus Volume 3,\u00a0 Calculus Volumes 1, 2, and 3 are licensed under an Attribution-NonCommercial-Sharealike 4.0 International License (CC BY-NC-SA).[\/footnote]<\/span>:\r\n\r\nCalculate the integrals by interchanging the order of integration:\r\n
      \r\n \t
    1. [latex]\\int_{-1}^{1} \\int_{-1}^{2} 2x + 3y + 5 \\, dx dy [\/latex]<\/li>\r\n \t
    2. [latex]\\int_{0}^{\\pi} \\int_{0}^{\\pi\/2}sin(2x) cos(3y)\\, dx dy [\/latex]<\/li>\r\n<\/ol>\r\nUse the graph to find the answers to these questions.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/header><\/header>","rendered":"

      The Concept<\/h2>\n

      The definition of the single integral<\/strong> in 2D space is as follows: given a single-variable function<\/strong> [latex]y=f(x)[\/latex] that is continuous on the interval [latex][a,b][\/latex], we divide the interval into [latex]n[\/latex] subintervals of equal width, [latex]x[\/latex], and from each interval choose a point, [latex]x_i[\/latex]. Definite integral,<\/strong> [latex]\\int_a^bf(x)dx[\/latex], represents the area inbetween the curve, [latex]y=f(x)[\/latex], and [latex]x[\/latex]-axis. Riemann sum<\/strong> helps to approximate such areas, that is,<\/p>\n

      [latex]\\int_a^bf(x)dx \\approx \\sum_{i=1}^n f(x_i) \\Delta x[\/latex],
      \nwhere [latex]\\Delta x = \\frac{b-a}{n}[\/latex] and [latex]x_i=a + i\\Delta x[\/latex]. The larger [latex]n[\/latex] is, the better the estimation is. Thus, the limit of the Riemann sum defines the definite integral,<\/p>\n

      [latex]\\int_a^bf(x)dx =\\lim_{n \\to \\infty} \\sum_{i=1}^n f(x_i) \\Delta x = \\sum_{i=1}^{\\infty} f(x_i)\\,\\Delta x[\/latex].<\/p>\n

       <\/p>\n

      Similar to the single integral, the double integral<\/strong> in 3D, [latex]\\iint_R f(x,y) dA[\/latex], is equal to the volume under the surface of the two-variable function<\/strong> [latex]z = f(x,y)[\/latex] and above the region [latex]R[\/latex] on the [latex]xy[\/latex]-plane. Here, we consider this region has a very simple shape, rectangle, and use [latex]R[\/latex] to denote it. The [latex]x[\/latex] coordinate of this rectangle changes from [latex]a[\/latex] to [latex]b[\/latex], and [latex]y[\/latex] coordinate changes from [latex]c[\/latex] to [latex]d[\/latex], denoted as [latex]R=[a, b]\\times[c,d][\/latex]. As in the case of the single integral, a double integral<\/strong> is defined as the limit of a Reimann sum<\/strong>, i.e.,<\/p>\n

      [latex]\\iint_R f(x,y) dA=\\lim_{m,n \\to \\infty}\\sum_{i=1}^{m} \\sum_{j=1}^{n} f(x_{i},y_{j}) \\Delta A[\/latex]<\/p>\n

      where [latex]\\Delta A=\\Delta x \\Delta y, \\Delta x=\\frac{b-a}{n}, \\Delta y=\\frac{d-c}{m}, x_i=a + i\\Delta x[\/latex] and [latex]y_i=c + i\\Delta y[\/latex].<\/p>\n

      The Plot<\/h2>\n

      Now, you should engage with the 3D plot below to understand double integrals over rectangular regions[1]<\/sup><\/a>. Follow the steps below to apply changes to the plot and observe the effects:<\/p>\n

        \n
      1. Input a function of two variables into the [latex]f(x,y)[\/latex] input function section.<\/li>\n
      2. Move the [latex]n[\/latex]-slide around to decide the subregions of the rectangular region, [latex]R[\/latex], and we consider the subregions are squares.<\/li>\n
      3. Pick xmin, xmax, ymin, and ymax points for your domain\/bounds of the rectangular region, [latex]R[\/latex].<\/li>\n
      4. Use the [latex]k[\/latex]-slider to choose which square-shaped subregion you\u2019d like to highlight.<\/li>\n
      5. Use the checkboxes to show either all of the rectangular prisms compared to just the one you are highlighting, as well as whether to see the graph or not.<\/li>\n
      6. By changing your view and hovering over the plot, you can see a 2D representation of the rectangular area. Additionally, the double integral is dynamically calculated at the bottom.<\/li>\n<\/ol>\n

        <\/p>\n

        \n
        <\/div>\n<\/div>\n

        <\/span><\/p>\n

        <\/header>\n
        <\/header>\n
        <\/header>\n
        \n
        \n
        \n

        Self-Checking Questions<\/h2>\n<\/header>\n
        \n

        Check your understanding by solving the following questions[2]<\/sup><\/a><\/span>:<\/p>\n

        Calculate the integrals by interchanging the order of integration:<\/p>\n

          \n
        1. [latex]\\int_{-1}^{1} \\int_{-1}^{2} 2x + 3y + 5 \\, dx dy[\/latex]<\/li>\n
        2. [latex]\\int_{0}^{\\pi} \\int_{0}^{\\pi\/2}sin(2x) cos(3y)\\, dx dy[\/latex]<\/li>\n<\/ol>\n

          Use the graph to find the answers to these questions.<\/p>\n<\/div>\n<\/div>\n<\/div>\n

          1. Made with GeoGebra, licensed Creative Commons CC BY-NC-SA 4.0. ↵<\/a><\/li>
          2. Gilbert Strang, Edwin \u201cJed\u201d Herman,\u00a0 OpenStax,\u00a0Calculus Volume 3,\u00a0 Calculus Volumes 1, 2, and 3 are licensed under an Attribution-NonCommercial-Sharealike 4.0 International License (CC BY-NC-SA). ↵<\/a><\/li><\/ol><\/div>","protected":false},"author":178,"menu_order":5,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-32","chapter","type-chapter","status-publish","hentry"],"part":3,"_links":{"self":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/wp-json\/pressbooks\/v2\/chapters\/32","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/wp-json\/wp\/v2\/users\/178"}],"version-history":[{"count":47,"href":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/wp-json\/pressbooks\/v2\/chapters\/32\/revisions"}],"predecessor-version":[{"id":546,"href":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/wp-json\/pressbooks\/v2\/chapters\/32\/revisions\/546"}],"part":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/wp-json\/pressbooks\/v2\/parts\/3"}],"metadata":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/wp-json\/pressbooks\/v2\/chapters\/32\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/wp-json\/wp\/v2\/media?parent=32"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=32"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/wp-json\/wp\/v2\/contributor?post=32"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/wp-json\/wp\/v2\/license?post=32"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}