{"id":134,"date":"2019-06-17T16:24:12","date_gmt":"2019-06-17T20:24:12","guid":{"rendered":"https:\/\/blogs.harvard.edu\/siams\/?p=134"},"modified":"2019-06-19T16:44:16","modified_gmt":"2019-06-19T20:44:16","slug":"hughes-hallett-et-al-chapter-8-using-the-definite-integral","status":"publish","type":"post","link":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/06\/17\/hughes-hallett-et-al-chapter-8-using-the-definite-integral\/","title":{"rendered":"Hughes-Hallett et al Chapter 8: Using the definite integral"},"content":{"rendered":"<p>For the course &#8220;Integrating and Approximating&#8221; our focus will be on multivariate integration, vector calculus, and differential equations. \u00a0In the past, I&#8217;ve used a number of texts for Multivariable, but appreciate the four-fold perspective (tables, graphs, formulas, words) that is used in Hughes-Hallett et al.<\/p>\n<p>A few chapters of single variable portion of the text are particularly relevant, so I&#8217;ll summarize them via blog posts.<\/p>\n<ul>\n<li>8.1 Areas and volumes\n<ul>\n<li>Find area via horizontal slides: Example 1 is a triangle, where horizontal slices are integrated to give the area. \u00a0Example 2 is a half-disk via horizontal slices. \u00a0Introducing (or reviewing) horizontal slices is a good idea before moving to 2d.<\/li>\n<li>Find volume via slices that are disks or squares: Example 3 is volume of a solid cone. \u00a0Vertical slices are a weird shape but horizontal ones are coins. \u00a0Example 4 is a half-ball via circular slices. \u00a0Example 5 is a pyramid, which has square slices.<\/li>\n<li>We could set up expressions for these areas using either single or double integrals, and expressions for these volumes using single, double, or triple integrals.<\/li>\n<\/ul>\n<\/li>\n<li><span style=\"font-size: 1rem\">8.2 Applications to geometry<\/span>\n<ul>\n<li>Find volume via slices that are disks or squares: Examples 1, 2, 3 are volumes made up of thickened disks or pieces of disks but that have more irregular shapes than above (think of a turned banister with varying radius). \u00a0Example 4 is an interesting and complicated shape where the cross sections are known to be squares.<\/li>\n<li>Find arclength: Examples 5 and 6 are arclength, including with a parametric curve in 2d.<\/li>\n<li>Arclength is worth doing in 2d (along with a parameterized curve in 2d) before returning to it in 3d: I can check whether students saw this in their Calc II course. \u00a0In addition, thinking through Example 4 would be worthwhile for working with the geometry of finding volumes.<\/li>\n<\/ul>\n<\/li>\n<li>8.3 Area and arc length in polar coordinates\n<ul>\n<li>Introduce polar coordinates, including their non-uniqueness: Example 1 is translating between polar and Cartesian coordinate systems. \u00a0Example 2 is giving different polar coordinates for a single point. \u00a0Examples 3 and 4 are about graphing a curve given in polar, and translating equations between polar and Cartesian.<\/li>\n<li>Introduce roses and limacons: Example 5 is two different roses and Example 6 is two different limacons.<\/li>\n<li>Use inequalities to describe regions. \u00a0Example 7 is describing an filled annulus in polar and Example 8 is describing a pizza slice.<\/li>\n<li>Find the area of a region described in polar by slicing into circular sectors: Example 9 is the area inside a limacon and Example 10 the area inside a petal of a rose graph.<\/li>\n<li>For a curve r = f(theta), find the slope. \u00a0Example 11 uses the formula for the slope of a parametric curve to find the slope.<\/li>\n<li>For a curve r = f(theta), find the arclength. \u00a0Example 12 is finding the arclength of one petal of a rose graph.<\/li>\n<li>This section has a lot in it. \u00a0The comprehensive intro to polar, including area, slope, and arclength, is worthwhile.<\/li>\n<\/ul>\n<\/li>\n<li>8.4 Density and center of mass\n<ul>\n<li>Finding a total quantity using a density function: Example 1 is population density along the Mass Turnpike. \u00a0Examples 2 and 3 are the mass of a solid-cylindrical column of air, and Example 4 is another population, but over a disk rather than along a line.<\/li>\n<li>Find the center of mass or balance point. \u00a0This involves defining displacement and moment. \u00a0Example 5 is a center of mass for children on a seesaw. \u00a0Example 6 works with a definition generalized to a number of point masses and Example 7 is with a continuous mass density.<\/li>\n<li>Find the center of mass for a 2d or 3d region. \u00a0Example 8 is an isosceles triangle and Example 9 a solid hemisphere.<\/li>\n<li>The definition of a density function, and practice with a density function is important.<\/li>\n<\/ul>\n<\/li>\n<li>8.5 Applications to physics\n<ul>\n<li>Work done by a force. \u00a0Example 1 is working with the definition. \u00a0Example 2 is the work to compress a spring. \u00a0Examples 4, 5, 6 are the work done by lifting a book, pumping oil to fill a tank, or building a pyramid.<\/li>\n<li>Use pressure to calculate force. \u00a0Example 7 is for a sunken ship and Example 8 is for the Hoover Dam.<\/li>\n<li>Pressure and work are both confusing topics in vector calculus so introducing them in the single variable context seems helpful.<\/li>\n<\/ul>\n<\/li>\n<li>8.6 Applications to economics\n<ul>\n<li>Present and future value of money: Example 1 is comparing a lump sum lottery payment vs installments. \u00a0Example 2 is looking at the value of an income stream.<\/li>\n<li>Supply and demand curves and consumer vs producer surplus (no associated examples)<\/li>\n<li>This doesn&#8217;t feed into any applications that I usually present in multivariable.<\/li>\n<\/ul>\n<\/li>\n<li>8.7 Distribution functions\n<ul>\n<li>Introduces a histogram (rather than using a count, they use one that is normalized, so that the total area is 1). \u00a0This leads to defining a probability density function. \u00a0Example 1 is estimating totals from a histogram, Example 2 is approximating a density function.<\/li>\n<li>The cumulative distribution is also defined (no associated examples)<\/li>\n<li>Introducing probability is worthwhile because most students take probability at some point, and it often isn&#8217;t introduced in a single variable course.<\/li>\n<\/ul>\n<\/li>\n<li>8.8 Probability, mean, and median\n<ul>\n<li>Find probability using a probability density function. \u00a0Example 1: use a pdf to identify a probability.<\/li>\n<li>Define median. \u00a0Example 2: use a pdf to find the median age of the US population.<\/li>\n<li>Define mean. \u00a0Example 3: use the pdf to find the mean age.<\/li>\n<li>Normal distribution. \u00a0Example 4: given a normal distribution with a given mean and stdev, find some probabilities.<\/li>\n<li>These are all worth defining, and doing them in 1d is probably simpler than in 2d.<\/li>\n<\/ul>\n<\/li>\n<li>Projects\n<ul>\n<li>Flux of fluid from a capillary<\/li>\n<li>Testing for kidney disease<\/li>\n<li>Volume enclosed by crossing cylinders<\/li>\n<li>Length of cable on the Golden Gate Bridge<\/li>\n<li>Surface area<\/li>\n<li>Maxwell&#8217;s distribution of molecular velocities<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>For the course &#8220;Integrating and Approximating&#8221; our focus will be on multivariate integration, vector calculus, and differential equations. \u00a0In the past, I&#8217;ve used a number of texts for Multivariable, but appreciate the four-fold perspective (tables, graphs, formulas, words) that is used in Hughes-Hallett et al. A few chapters of single variable portion of the text [&hellip;]<\/p>\n","protected":false},"author":8032,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[1010,157885],"tags":[],"class_list":["post-134","post","type-post","status-publish","format-standard","hentry","category-math","category-multivariable"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/p7E5LF-2a","jetpack-related-posts":[{"id":141,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/06\/19\/hughes-hallett-et-al-chapter-11-differential-equations\/","url_meta":{"origin":134,"position":0},"title":"Hughes-Hallett et al Chapter 11: Differential equations","author":"siams","date":"19 June 2019","format":false,"excerpt":"11.1: What is a differential equation? Starts with an example: what sets the rate at which a person learns a new task? \u00a0Defines a diff eq and a solution to a diff eq. Defines order of a diff eq. \u00a0Example 1 is showing a function is not a solution to\u2026","rel":"","context":"In &quot;Math&quot;","block_context":{"text":"Math","link":"https:\/\/archive.blogs.harvard.edu\/siams\/category\/math\/"},"img":{"alt_text":"","src":"","width":0,"height":0},"classes":[]},{"id":193,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/07\/22\/notes-on-calculus-blue-volume-1-chapter-3-and-4\/","url_meta":{"origin":134,"position":1},"title":"Notes on Calculus Blue Volume 1, Chapters 3, 4, 5, 6","author":"siams","date":"22 July 2019","format":false,"excerpt":"More of Calculus Blue by Prof Ghrist Math. Chapter 3 is on coordinates and distance (see section 12.1 of the 6th edition of Hughes-Hallett). 01.03 (0:36) \"Coordinates: intro\". \u00a0Review coordinates and see it in data. 01.03.01 (2:16) \"coordinates & many dimensions\". \u00a0from curves and surfaces we'll head on. \u00a0plane, then\u2026","rel":"","context":"In &quot;Math&quot;","block_context":{"text":"Math","link":"https:\/\/archive.blogs.harvard.edu\/siams\/category\/math\/"},"img":{"alt_text":"","src":"","width":0,"height":0},"classes":[]},{"id":65,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2017\/10\/04\/other-versions-of-multivariable\/","url_meta":{"origin":134,"position":2},"title":"Other versions of multivariable","author":"siams","date":"4 October 2017","format":false,"excerpt":"Resources for multivariable calculus: Some challenge problems (not multivariable): https:\/\/www.math.unl.edu\/~mrammaha1\/Challenging%20problems\/Challenge-Problems.pdf Cornell is using workshops activities for their engineering students this year (2017): http:\/\/www.math.cornell.edu\/~web1920\/workshop.html Materials from Math 53 at Berkeley (2016): https:\/\/math.berkeley.edu\/~auroux\/53s16\/ Lots of past multivariable exams for Math 215 at Michigan: http:\/\/www.math.lsa.umich.edu\/courses\/215\/17exampractice\/index.html","rel":"","context":"In &quot;Math&quot;","block_context":{"text":"Math","link":"https:\/\/archive.blogs.harvard.edu\/siams\/category\/math\/"},"img":{"alt_text":"","src":"","width":0,"height":0},"classes":[]},{"id":186,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/07\/22\/notes-on-calculus-blue-volume-1-chapter-1\/","url_meta":{"origin":134,"position":3},"title":"Notes on &#8220;Calculus Blue&#8221; Volume 1, Chapter 1","author":"siams","date":"22 July 2019","format":false,"excerpt":"These notes are on the Calculus Blue videos by Ghrist on YouTube. \u00a0He emphasizes that the math will involve substantial (and worthwhile) work, which I really appreciate. 01 (0:51) \"Vectors & matrices: Intro\" \u00a0\"Your journey is not a short one\". \u00a0To learn \"calculus, the mathematics of the nonlinear\", prepare with\u2026","rel":"","context":"In &quot;Math&quot;","block_context":{"text":"Math","link":"https:\/\/archive.blogs.harvard.edu\/siams\/category\/math\/"},"img":{"alt_text":"","src":"","width":0,"height":0},"classes":[]},{"id":188,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/07\/22\/notes-on-calculus-blue-volume-1-chapter-2\/","url_meta":{"origin":134,"position":4},"title":"Notes on &#8220;Calculus Blue&#8221; Volume 1, Chapter 2","author":"siams","date":"22 July 2019","format":false,"excerpt":"More notes on the Calculus Blue Multivariable Volume 1 videos on YouTube by \"Prof Ghrist Math\". Chapter 2 introduces curves in the plane and surfaces in 3-space with implicit and parametric definitions for curves in the plane and for surfaces in 3-space. \u00a0They also introduce the names and images for\u2026","rel":"","context":"In &quot;Math&quot;","block_context":{"text":"Math","link":"https:\/\/archive.blogs.harvard.edu\/siams\/category\/math\/"},"img":{"alt_text":"","src":"","width":0,"height":0},"classes":[]},{"id":58,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2017\/08\/25\/vector-calculus-earlier-in-the-semester\/","url_meta":{"origin":134,"position":5},"title":"Vector calculus earlier in the semester?","author":"siams","date":"25 August 2017","format":false,"excerpt":"Flux is a particularly central scientific and mathematically idea that appears in the context of a multivariable calculus course. \u00a0Given a velocity vector field and a surface, the flux of the vector field through the surface tells us the rate at which fluid is flowing through the surface. \u00a0This leads\u2026","rel":"","context":"In &quot;Math&quot;","block_context":{"text":"Math","link":"https:\/\/archive.blogs.harvard.edu\/siams\/category\/math\/"},"img":{"alt_text":"","src":"","width":0,"height":0},"classes":[]}],"_links":{"self":[{"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/posts\/134","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/users\/8032"}],"replies":[{"embeddable":true,"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/comments?post=134"}],"version-history":[{"count":6,"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/posts\/134\/revisions"}],"predecessor-version":[{"id":143,"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/posts\/134\/revisions\/143"}],"wp:attachment":[{"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/media?parent=134"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/categories?post=134"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/tags?post=134"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}