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Calculus & Mathematica: 152 Course Description

Integrals: Measurements of Accumulated Growths

2.01 Integrals for Measuring Area

Mathematics

Integrals defined as area measurement as done in E. Artin's MAA notes written in the 1950's. Approximations by trapezoids.

Science and Math Experience

Integrals of functions given by data lists. Using known area formulas for triangles, trapezoids and circles to calculate integrals. Odd functions. Trying to break the code of the integral by taking selected functions g[x], putting f[t] = g[x] dx and plotting (f[t + h] - f[t])/h and g[t] on the same axes for small h's. Plotting f[x] = cos[t] dt and guessing a formula for f[x]. Plotting f[x] = sin[t] dt and guessing a formula for f[x]. Estimating the acreage of farm field bordered by a river.

2.02 Breaking the Code of the Integral: The Fundamental Formula

Mathematics

If f[t] is given by f[t] = g[x] dx, then f'[t] = g[t]. The fundamental formula f[t] - f[a] = f'[x[ dx.

Science and Math Experience

Relating distance, velocity and acceleration through the fundamental formula. Getting the feel of the fundamental formula by using it to calculate integrals by hand. Relating g[x] dx to the solution of the differential equation y'[t] = g[t] with y[a] = 0. Very brief look at the "indefinite integral," g[x] dx. Measuring area between curves. The error function, erf[x], and other functions defined by integrals. Measurements of accumulated growth. Coloring ceramic tiles. 

2.03 Measurements

Mathematics

Measurements based on slicing and accumulating: Area and volume; density and mass. Measurements based on approximating and measuring: Arc length. Measurements based on the fundamental formula: Accumulated growth.

Science and Math Experience

Volumes of solids with no special emphasis on solids of rotation. Volume measurements of curved tubes and horns. Eyeball and precise estimates of curve lengths. Filling water tanks. Harvesting corn. Voltage drop. Another look at linear dimension. Work. Present value of a profit-making scheme. Catfish harvesting. Designing an 8 fluid ounce logarithmic champagne glass.

2.04 Transforming Integrals

Mathematics

Using the chain rule and the fundamental formula to see why f'[u[x]] u'[x] dx = and using this fact to transform one integral into another. Measuring area under curves given parametrically. Bell shaped curves and Gauss's normal probability law; mean and standard deviation.

Science and Math Experience

Study of the error function, erf[x]. Using transformations to explain Mathematica output. Polar plots and area measurements. Using transformations to explain the meaning of standard deviation in Gauss's normal law. Expected life of light bulbs and how long to set the guarantee on them. Using Gauss's normal law to help to program coin-operated coffee machines. IQ test results. Using Gauss's normal law to organize SAT scores into quartiles and deciles. Comparison of 1967 and 1987 SAT scores. "Grading on the curve."

2.05 2D Integrals

Mathematics

Meaning of the plot of z = f[x,y]. The 2D integral _R f[x,y] dx dy as a volume measurement via slicing and accumulating. Gauss-Green formula (Green's theorem) as a way of calculating a double integral numerically as a single integral.

Science and Math Experience

Volume and area measurements with 2D integrals. Area and volume measurements via the Gauss-Green formula. Average value and centroids. Calculation strategies. Plotting and measuring. Gauss's normal law in 2D and using it, as done in the Pentagon, to decide how many bombs to drop on a target.

2.06 More Tools and Measurements

Mathematics

Separating the variables and integrating to get formulas for the solutions of some differential equations. Integration by parts. Complex numbers and the complex exponential e^a+ib = e^a (Cos[b] + I Sin[b]),

Science and Math Experience

Formulas for the solutions of the differential equations involved in the chemical model and the spread of infection model. Hyperbolic functions and their relation to trigonometric functions. Using the complex exponential to help to understand the Mathematica output from the Solve instruction. Gamma function. Integration by parts and integration by iteration. Error propagation in forward iteration. Error reduction by backward iteration.

2.07 Traditional Pat Integration Procedures for Special Situations

Mathematics

Undetermined coefficients. Complex numbers and partial fractions. Wild card substitutions with the help of a trigonometric, hyperbolic or ad hoc function. Integration by parts.

Science and Math Experience

Not much, although the experience gained from trying the method of undetermined coefficients is good experience in setting up and solving systems of linear equations.

Significant Changes From the Traditional Course

• Student writing, plotting and experimentation is the stock in trade of the course.
• Following Courant and Artin, the integral is introduced as a measurement of area under a curve. The approach is strong enough to get to the idea of approximating by trapezoids and to get to the idea of the fundamental formula while totally avoiding the bureaucracy of Riemann sums. This results in significant saving of time for more important matters.
• Riemann sums are used only occasionally and they are used only as part of the process of setting up an integral for a required measurement.
• Integrals are used to measure area, length and volume but are not used to define these notions. The issue of defining these notions is not addressed.
• The indefinite integral, f[x] dx, (the integral without limits) is totally de-emphasized and mentioned only in passing. Reason: In this course, all integrals make measurements; the indefinite integral measures nothing. On the other hand, the functions like f[x] dx (as a function of t) are under heavy scrutiny.
• The idea for the fundamental formula is set up experimentally by having the student take selected functions g[x], putting f[t] = g[x] dx and plotting (f[t + h] - f[t])/h and g[t] on the same axes for small h's. This experiment not only sets up the fundamental formula but also reinforces the idea of what the derivative is.
• The explanation of the fundamental formula is based on the trapezoidal rule.
• Threefold emphasis on measurements made with integrals: Measurements based on slicing and accumulating, measurements based on approximations and measurements of accumulated growth via the fundamental formula.
• At the request of physics professors, the complex exponential enters an American calculus course for the first time. The matter-of-fact treatment is based in part on ideas from the Feynman Lectures.
• Although traditional techniques of integration are de-emphasized, integration by parts and the idea of transforming one integral into another via the formula f'[u[x]] u'[x] dx = f'[u] du survive. Reason: Knowledge of transformations of integrals helps to prepare the student to understand where Mathematica-generated formulas like Sin[Pi/2 t²] dt = FresnelS[x] come from and what they mean.
• The second reason for emphasis on transforming integrals is that simple transformations unlock the mysteries of the normal probability distribution. Most students have heard of the famous "bell-shaped curve" and want to work with it. In this course, they get their chance by using properties of the normal probability distribution to set quartiles on SAT scores and they even have the experience of "curving a test."
• Double integrals as volume measurements are taken up earlier than they are in the traditional course. Mathematica's ability to plot surfaces is a strong motivator for taking up this topic as soon as possible. As a calculational aide, Gauss-Green formula (Green's theorem) is introduced for the purpose of replacing calculationally intractable double integrals by one dimensional integrals which can be easily calculated numerically. Students get to try this out by using Gauss's normal law in 2D to decide, as done in the Pentagon, how many bombs to drop on a target to get a given kill probability. Student will get another chance to work with the Gauss-Green formula in vector calculus. Students appreciate the chance to have two cracks at the Gauss-Green formula.