1
Compute total change from a rate
Cut an interval into equal pieces, add rate times width exactly, and state what the sum assumed about the rate inside each piece.
Course · Differential Equations
Accumulation and the Integral
An integral is accumulated change. This course adds up a rate piece by piece in exact fractions, watches the sums close in on a number, and then shows that reversing the power rule finds that number without any sum. The one constant that reversal leaves open is the reason an equation later needs an initial value.
total change ≈ Σ (rate · width)
Rₙ − Lₙ = h·(f(b) − f(a))
Tₙ = (Lₙ + Rₙ)/2, Mₙ from midpoint rates
∫ₐᵇ f(t) dt = F(b) − F(a)
y = F(t) + C one value fixes C
∫₁ˣ dt/t = ln x not a fraction
What this course is about
- Subject
- Differential Equations
- Lessons
- 8
- Format
- Interactive
Overview
What you will be able to do
By the end you can estimate a total change from a rate with exact sums, bracket it, find it by an antiderivative, and say what each of those claims and what it only demonstrates.
2
Bracket and refine a total
Compute left and right sums, predict their gap, and double the pieces to watch the error and its exact ratio.
3
Compare four rules
Compute trapezoid and midpoint sums, find their exact errors, and show the error quartering on a quadratic.
4
Find a total by antiderivative
Reverse the power rule, evaluate F(b) − F(a) exactly, and fix the constant from one given value.
5
Read and use the integral sign
Say ∫ₐᵇ f(t) dt aloud, and check linearity and additivity by computing both sides exactly.
6
Bracket an integral that is not a fraction
Sum 1/t exactly, trap the total between the left and right sums, and quote it rounded and labelled rather than as a fraction.
Lessons
8 lessons
Adding up a rate comes first: sums, their gap, their refinement and the better rules. The exact total follows, by antiderivative and by a constant fixed from one value. The last two lessons give the notation and the one integral that no fraction equals.
Total Change from a RateTotal change is rate times width added across pieces, and the sum is only as good as the assumption that the rate holds still inside each piece.Not marked
Left and Right SumsThe left and right sums differ by exactly h times the change in the rate across the interval, and for a monotone rate the total lies between them.Not marked
Refining the PartitionDoubling the number of pieces moves the exact sums toward a single number, and the table shows that without proving it.Not marked
The Trapezoid and Midpoint RulesAveraging the left and right sums, or reading the rate at the middle of each piece, makes the error fall by about four when the pieces double, and by exactly four on a quadratic.Not marked
The Antiderivative and the Fundamental TheoremAn antiderivative of a rate gives the total change as its value at the right end minus its value at the left, with no pieces to add.Not marked
The Constant of IntegrationA rate has a whole family of antiderivatives, each differing by a constant, and exactly one given value picks the member.Not marked
The Integral Sign and Its RulesThe integral sign names the total change a rate produces over an interval, it obeys linearity and additivity, and the integral of a product is not the product of the integrals.Not marked
The Integral of 1/tThe reverse power rule has no answer for one over t, and the exact sums bracket a number that is not a fraction, written ln x.Not marked
Recommended background
The derivative and the power rule from Rates of Change and the Derivative, and fractions.
Practice suggestions
Compute the first sum of each lesson by hand before reading the lab's figure. The sums are short, and doing one yourself makes the table a check and not a surprise.
Keep what a table shows apart from what it proves. Several lessons here state a claim, demonstrate it on exact fractions, and say plainly that the demonstration is not a proof for every case.
After each worked example, cover it and finish the faded rehearsal beneath it before opening the quiz. Change the rate in the lab and predict the sign of the error before you look.
What it does not cover
The theory of the integral. There is no definition by limits of arbitrary partitions, no test for which rates have an integral, and no proof that the sums converge; the sums are shown approaching a number and the fundamental theorem is stated as the claim they support. A first analysis course is where that is done.
Techniques of integration for general functions. Integration by parts, substitution, trigonometric substitution and integrating rational functions by partial fractions are left out; the Subject needs polynomials and 1/t and nothing else.
Improper integrals, integrals over infinite intervals, and areas between curves. Totals here are signed accumulations of a rate over a finite interval.
Numerical integration as a subject. Adaptive step size, Simpson's rule and error estimators are not here; the four rules and their exact errors on polynomials are the whole of it.