{"id":246,"date":"2022-02-23T13:13:57","date_gmt":"2022-02-23T18:13:57","guid":{"rendered":"https:\/\/pressbooks.library.ryerson.ca\/multivariatecalculus\/?post_type=chapter&p=246"},"modified":"2022-03-03T22:42:34","modified_gmt":"2022-03-04T03:42:34","slug":"unit-10-3d-solid-bounded-by-two-surfaces","status":"publish","type":"chapter","link":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/chapter\/unit-10-3d-solid-bounded-by-two-surfaces\/","title":{"raw":"Unit 10: 3D Solid Bounded by Two Surfaces","rendered":"Unit 10: 3D Solid Bounded by Two Surfaces"},"content":{"raw":"

The Concept\u00a0<\/strong><\/h2>\r\nThe graphs of functions of two variables<\/strong>\u00a0[latex]z=f(x, y)[\/latex] are examples of surfaces in 3D. More generally, a set of points [latex](x,y,z)[\/latex] that satisfy an equation relating all three variables is often a surface. A simple example is the unit sphere, the set of points that satisfy the equation [latex]x^2+y^2+z^2=1[\/latex].\r\n\r\nOne special class of equations is a set of equations that involve one or more [latex]x^2, y^2, z^2, xy, xz[\/latex], and [latex]yz[\/latex]. The graphs of these equations are surfaces known as quadric surfaces<\/strong>. There are six different quadric surfaces: the ellipsoid, the elliptic paraboloid, the hyperbolic paraboloid, the double cone and hyperboloids of one sheet and two sheets. Quadric surfaces are natural 3D extensions of the so-called conics (ellipses, parabolas and hyperbolas), and they provide examples of fairly nice surfaces to use as examples in multivariate calculus.\r\n

The Plot<\/h2>\r\nNow, you should engage with the 3D plot below to understand 3D solids bounded by two surfaces[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. Fill in function 1 (i.e., [latex]f(x,y,z)[\/latex]) and function 2 (i.e., [latex]g(x,y,z)[\/latex]) with your desired quadric surfaces.<\/li>\r\n \t
  2. The graph depicted on the right shows their intersection.<\/li>\r\n<\/ol>\r\n[h5p id=\"22\"]<\/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\nPlot the given quadric surface and specify the name of said quadric surface:\r\n
      \r\n \t
    1. [latex]x^2\/4 + y^2\/9 - z^2\/12 = 1[\/latex]<\/li>\r\n \t
    2. [latex]z^2 = 4x^2 + 3y^2[\/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 ","rendered":"

      The Concept\u00a0<\/strong><\/h2>\n

      The graphs of functions of two variables<\/strong>\u00a0[latex]z=f(x, y)[\/latex] are examples of surfaces in 3D. More generally, a set of points [latex](x,y,z)[\/latex] that satisfy an equation relating all three variables is often a surface. A simple example is the unit sphere, the set of points that satisfy the equation [latex]x^2+y^2+z^2=1[\/latex].<\/p>\n

      One special class of equations is a set of equations that involve one or more [latex]x^2, y^2, z^2, xy, xz[\/latex], and [latex]yz[\/latex]. The graphs of these equations are surfaces known as quadric surfaces<\/strong>. There are six different quadric surfaces: the ellipsoid, the elliptic paraboloid, the hyperbolic paraboloid, the double cone and hyperboloids of one sheet and two sheets. Quadric surfaces are natural 3D extensions of the so-called conics (ellipses, parabolas and hyperbolas), and they provide examples of fairly nice surfaces to use as examples in multivariate calculus.<\/p>\n

      The Plot<\/h2>\n

      Now, you should engage with the 3D plot below to understand 3D solids bounded by two surfaces[1]<\/sup><\/a>. Follow the steps below to apply changes to the plot and observe the effects:<\/p>\n

        \n
      1. Fill in function 1 (i.e., [latex]f(x,y,z)[\/latex]) and function 2 (i.e., [latex]g(x,y,z)[\/latex]) with your desired quadric surfaces.<\/li>\n
      2. The graph depicted on the right shows their intersection.<\/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

        Plot the given quadric surface and specify the name of said quadric surface:<\/p>\n

          \n
        1. [latex]x^2\/4 + y^2\/9 - z^2\/12 = 1[\/latex]<\/li>\n
        2. [latex]z^2 = 4x^2 + 3y^2[\/latex]<\/li>\n<\/ol>\n

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

           <\/p>\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":10,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-246","chapter","type-chapter","status-publish","hentry"],"part":3,"_links":{"self":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/wp-json\/pressbooks\/v2\/chapters\/246","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":10,"href":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/wp-json\/pressbooks\/v2\/chapters\/246\/revisions"}],"predecessor-version":[{"id":551,"href":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/wp-json\/pressbooks\/v2\/chapters\/246\/revisions\/551"}],"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\/246\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/wp-json\/wp\/v2\/media?parent=246"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=246"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/wp-json\/wp\/v2\/contributor?post=246"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/wp-json\/wp\/v2\/license?post=246"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}