Hughes-Hallett et al Chapter 8: Using the definite integral

For the course “Integrating and Approximating” our focus will be on multivariate integration, vector calculus, and differential equations.  In the past, I’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 are particularly relevant, so I’ll summarize them via blog posts.

  • 8.1 Areas and volumes
    • Find area via horizontal slides: Example 1 is a triangle, where horizontal slices are integrated to give the area.  Example 2 is a half-disk via horizontal slices.  Introducing (or reviewing) horizontal slices is a good idea before moving to 2d.
    • Find volume via slices that are disks or squares: Example 3 is volume of a solid cone.  Vertical slices are a weird shape but horizontal ones are coins.  Example 4 is a half-ball via circular slices.  Example 5 is a pyramid, which has square slices.
    • 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.
  • 8.2 Applications to geometry
    • 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).  Example 4 is an interesting and complicated shape where the cross sections are known to be squares.
    • Find arclength: Examples 5 and 6 are arclength, including with a parametric curve in 2d.
    • 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.  In addition, thinking through Example 4 would be worthwhile for working with the geometry of finding volumes.
  • 8.3 Area and arc length in polar coordinates
    • Introduce polar coordinates, including their non-uniqueness: Example 1 is translating between polar and Cartesian coordinate systems.  Example 2 is giving different polar coordinates for a single point.  Examples 3 and 4 are about graphing a curve given in polar, and translating equations between polar and Cartesian.
    • Introduce roses and limacons: Example 5 is two different roses and Example 6 is two different limacons.
    • Use inequalities to describe regions.  Example 7 is describing an filled annulus in polar and Example 8 is describing a pizza slice.
    • 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.
    • For a curve r = f(theta), find the slope.  Example 11 uses the formula for the slope of a parametric curve to find the slope.
    • For a curve r = f(theta), find the arclength.  Example 12 is finding the arclength of one petal of a rose graph.
    • This section has a lot in it.  The comprehensive intro to polar, including area, slope, and arclength, is worthwhile.
  • 8.4 Density and center of mass
    • Finding a total quantity using a density function: Example 1 is population density along the Mass Turnpike.  Examples 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.
    • Find the center of mass or balance point.  This involves defining displacement and moment.  Example 5 is a center of mass for children on a seesaw.  Example 6 works with a definition generalized to a number of point masses and Example 7 is with a continuous mass density.
    • Find the center of mass for a 2d or 3d region.  Example 8 is an isosceles triangle and Example 9 a solid hemisphere.
    • The definition of a density function, and practice with a density function is important.
  • 8.5 Applications to physics
    • Work done by a force.  Example 1 is working with the definition.  Example 2 is the work to compress a spring.  Examples 4, 5, 6 are the work done by lifting a book, pumping oil to fill a tank, or building a pyramid.
    • Use pressure to calculate force.  Example 7 is for a sunken ship and Example 8 is for the Hoover Dam.
    • Pressure and work are both confusing topics in vector calculus so introducing them in the single variable context seems helpful.
  • 8.6 Applications to economics
    • Present and future value of money: Example 1 is comparing a lump sum lottery payment vs installments.  Example 2 is looking at the value of an income stream.
    • Supply and demand curves and consumer vs producer surplus (no associated examples)
    • This doesn’t feed into any applications that I usually present in multivariable.
  • 8.7 Distribution functions
    • Introduces a histogram (rather than using a count, they use one that is normalized, so that the total area is 1).  This leads to defining a probability density function.  Example 1 is estimating totals from a histogram, Example 2 is approximating a density function.
    • The cumulative distribution is also defined (no associated examples)
    • Introducing probability is worthwhile because most students take probability at some point, and it often isn’t introduced in a single variable course.
  • 8.8 Probability, mean, and median
    • Find probability using a probability density function.  Example 1: use a pdf to identify a probability.
    • Define median.  Example 2: use a pdf to find the median age of the US population.
    • Define mean.  Example 3: use the pdf to find the mean age.
    • Normal distribution.  Example 4: given a normal distribution with a given mean and stdev, find some probabilities.
    • These are all worth defining, and doing them in 1d is probably simpler than in 2d.
  • Projects
    • Flux of fluid from a capillary
    • Testing for kidney disease
    • Volume enclosed by crossing cylinders
    • Length of cable on the Golden Gate Bridge
    • Surface area
    • Maxwell’s distribution of molecular velocities