# Calculus Book 2: Accumulation

A second book in Calculus. Integrals for Measuring Area, the Fundamental Formula, Measurements, Transforming Integrals, 2-D Integrals, Gauss-Green Formula, Integration Procedures, and more!

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**Syllabus**

**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[x]=g[t]dt and plotting (f[x + h] – f[x])/h and g[x] 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 The Fundamental Formula**

*Mathematics.*

If a function f is defined by f[x]=g[t]dt then f'[x]=g[x].

The fundamental formula: f[x]-f[a]=f'[t]dt.

*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[t]dt to the solution of the differential equation y'[x]=g[x] with y[a]=0.

Very brief look at the “indefinite integral,” ∫g[t]dt

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 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 and the Gauss-Green Formula**

*Mathematics.*

Meaning of the plot of z=f[x,y]. The 2D integral 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.

*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 backwards iteration.

**2.07 Traditional Pat Integration Procedures**

The nasty quotient simplifies to .

So the nasty quotient is easy to integrate.

*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.