{"id":181,"date":"2022-02-22T22:46:08","date_gmt":"2022-02-23T03:46:08","guid":{"rendered":"https:\/\/pressbooks.library.ryerson.ca\/multivariatecalculus\/?post_type=chapter&p=181"},"modified":"2024-01-30T22:36:02","modified_gmt":"2024-01-31T03:36:02","slug":"unit-7-double-integrals-in-polar-coordinates","status":"publish","type":"chapter","link":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/chapter\/unit-7-double-integrals-in-polar-coordinates\/","title":{"raw":"Unit 7: Double Integrals in Polar Coordinates","rendered":"Unit 7: Double Integrals in Polar Coordinates"},"content":{"raw":"

The Concept<\/h2>\r\nNow we will look at the concept of double integrals in polar coordinates<\/strong>. Rather than using a cartesian (or rectangular) coordinate system<\/strong> as we have used thus far to evaluate single and double integrals, we will use the polar coordinate system. The polar coordinate system<\/strong> is a 2D coordinate system in which each point on a plane is determined using a distance from a reference point and an angle from a reference direction. The rectangular coordinate system is best suited for graphs and regions that are naturally considered over a rectangular grid. The polar coordinate system is an alternative that offers good options for functions and domains that have more circular characteristics.\r\n\r\nWhile a point [latex]P[\/latex] in rectangular coordinates is described by an ordered pair [latex](x,y)[\/latex], it may also be described in polar coordinates by [latex](r, \\theta)[\/latex], where r is the distance from [latex]P[\/latex] to the origin and [latex]\\theta[\/latex] is the angle formed by the line segment and the positive [latex]x[\/latex]x-axis. We may convert a point from rectangular to polar coordinates using the following equations:\r\n

[latex]r =\\sqrt{x^2+y^2}[\/latex]\u00a0 and [latex]\\tan(\\theta) = \\frac{y}{x}[\/latex],<\/p>\r\nor convert a point from polar to rectangular coordinates using the following equations:\r\n

[latex]x =r \\cos\\theta[\/latex]\u00a0 and [latex]y = r \\sin\\theta[\/latex].<\/p>\r\nThe double integral [latex]\\iint_D f(x,y)\\,dA[\/latex] in rectangular coordinates can be converted to a double integral in polar coordinates as [latex]\\iint_D f(r \\cos\\theta, r \\sin\\theta)\\,r\\,dr d\\theta[\/latex].\r\n

The Plot<\/h2>\r\nNow, you should engage with the plot below to understand polar coordinates[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. Change the bounds on the double integral in polar coordinates for both the [latex]r[\/latex] and [latex]\\theta[\/latex] bounds. The bounded region will be shown in the plot and [latex]t[\/latex] in the plot represents [latex]\\theta[\/latex].<\/li>\r\n \t
  2. The result of the double integral in polar coordinates will be shown too.<\/li>\r\n<\/ol>\r\n[h5p id=\"19\"]<\/span>\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\n \t
    1. [latex]\\iint_{D}\u00a0 3x \\, dA [\/latex] where [latex]R=\\{(r,\\theta)| 0 \\leq r \\leq 1, 0 \\leq \\theta \\leq 2\\pi\\}[\/latex].<\/li>\r\n \t
    2. [latex]\\iint_{D}\u00a0 1-x^2-y^2 \\, dA [\/latex] where [latex]R=\\{(r,\\theta)| 0 \\leq r \\leq 1, 0 \\leq \\theta \\leq 2\\pi\\}[\/latex].<\/li>\r\n<\/ol>\r\nUse the graph to find the answers to these questions.\r\n\r\n<\/div>\r\n<\/div>","rendered":"

      The Concept<\/h2>\n

      Now we will look at the concept of double integrals in polar coordinates<\/strong>. Rather than using a cartesian (or rectangular) coordinate system<\/strong> as we have used thus far to evaluate single and double integrals, we will use the polar coordinate system. The polar coordinate system<\/strong> is a 2D coordinate system in which each point on a plane is determined using a distance from a reference point and an angle from a reference direction. The rectangular coordinate system is best suited for graphs and regions that are naturally considered over a rectangular grid. The polar coordinate system is an alternative that offers good options for functions and domains that have more circular characteristics.<\/p>\n

      While a point [latex]P[\/latex] in rectangular coordinates is described by an ordered pair [latex](x,y)[\/latex], it may also be described in polar coordinates by [latex](r, \\theta)[\/latex], where r is the distance from [latex]P[\/latex] to the origin and [latex]\\theta[\/latex] is the angle formed by the line segment and the positive [latex]x[\/latex]x-axis. We may convert a point from rectangular to polar coordinates using the following equations:<\/p>\n

      [latex]r =\\sqrt{x^2+y^2}[\/latex]\u00a0 and [latex]\\tan(\\theta) = \\frac{y}{x}[\/latex],<\/p>\n

      or convert a point from polar to rectangular coordinates using the following equations:<\/p>\n

      [latex]x =r \\cos\\theta[\/latex]\u00a0 and [latex]y = r \\sin\\theta[\/latex].<\/p>\n

      The double integral [latex]\\iint_D f(x,y)\\,dA[\/latex] in rectangular coordinates can be converted to a double integral in polar coordinates as [latex]\\iint_D f(r \\cos\\theta, r \\sin\\theta)\\,r\\,dr d\\theta[\/latex].<\/p>\n

      The Plot<\/h2>\n

      Now, you should engage with the plot below to understand polar coordinates[1]<\/sup><\/a>. Follow the steps below to apply changes to the plot and observe the effects:<\/p>\n

        \n
      1. Change the bounds on the double integral in polar coordinates for both the [latex]r[\/latex] and [latex]\\theta[\/latex] bounds. The bounded region will be shown in the plot and [latex]t[\/latex] in the plot represents [latex]\\theta[\/latex].<\/li>\n
      2. The result of the double integral in polar coordinates will be shown too.<\/li>\n<\/ol>\n

        <\/p>\n

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

        <\/span><\/p>\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

          \n
        1. [latex]\\iint_{D}\u00a0 3x \\, dA[\/latex] where [latex]R=\\{(r,\\theta)| 0 \\leq r \\leq 1, 0 \\leq \\theta \\leq 2\\pi\\}[\/latex].<\/li>\n
        2. [latex]\\iint_{D}\u00a0 1-x^2-y^2 \\, dA[\/latex] where [latex]R=\\{(r,\\theta)| 0 \\leq r \\leq 1, 0 \\leq \\theta \\leq 2\\pi\\}[\/latex].<\/li>\n<\/ol>\n

          Use the graph to find the answers to these questions.<\/p>\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":391,"menu_order":7,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-181","chapter","type-chapter","status-publish","hentry"],"part":3,"_links":{"self":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/wp-json\/pressbooks\/v2\/chapters\/181","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\/391"}],"version-history":[{"count":23,"href":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/wp-json\/pressbooks\/v2\/chapters\/181\/revisions"}],"predecessor-version":[{"id":641,"href":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/wp-json\/pressbooks\/v2\/chapters\/181\/revisions\/641"}],"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\/181\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/wp-json\/wp\/v2\/media?parent=181"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=181"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/wp-json\/wp\/v2\/contributor?post=181"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/wp-json\/wp\/v2\/license?post=181"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}