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Free lesson · Integral calculus

Start with rectangles: rate × time = amount

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Start with the idea

  • An integral adds small contributions.
  • Begin with a constant rate, where the total is a rectangle.
Symbols, units & horizon
  • t: time in minutes
  • a,b: start and end times in minutes, b≥a in this example
  • c: constant flow in litres per minute
  • ∫: accumulate over the stated interval
  • dt: integration with respect to time
  • n: positive number of equal subintervals

When and why to use this

Use accumulation to combine a rate over time. A constant execution rate gives shares traded; a changing rate needs the later sum and integral rules.

  • Rate: amount per unit time.
  • Duration: how long the rate operates.
  • Multiply rate by duration to get the accumulated amount.
  • The integral sign ∫ means continuous accumulation.
  • The lower and upper limits mark the starting and ending inputs.
  • The symbol dt identifies time as the variable of integration.
∫abcdt=c(b−a)
Core rule · definition and worked arithmetic

Start with rectangles: rate × time = amount

  1. Split the interval into n equal widths: (b−a)n.
  2. Each piece contributes c(b−a)n litres.
  3. Add n identical pieces: nc(b−a)n=c(b−a).
  4. Every subdivision gives the same total; taking finer subdivisions preserves it.
Work it by hand

Flow c=3 litres/minute, from minute 1 to minute 5: duration=4 minutes, total=3×4=12 litres.

Use the rule

  • Name the inputs and units.
  • Work the small example by hand.
  • Check the result before continuing to the next lesson.

Before moving on

Explain the core rule in one sentence, reproduce the worked calculation and solve both practice variations.

Further reading: OpenStax · Definite integrals · textbook checked 12 September 2026 ↗

Research sources, review dates and limitations

Python implementation

Self-contained teaching example. Python 3.10+; dependencies and input conventions are shown in the code and notation. Run in your own Python environment.

def constant_integral(rate, start, end):
    return rate*(end-start)

print(constant_integral(3,1,5))

Continue learning

Integral Calculus: Accumulation & Area — all lessons
  1. Start with rectangles: rate × time = amount
  2. Definite integrals: adding many small contributions
  3. Core integral rules: undo a derivative, then check
  4. Antiderivatives and the fundamental theorem
  5. Integration techniques: substitution and integration by parts
  6. Improper integrals, densities and continuous expectations
  7. Numerical integration and a continuous cash-flow building block

Quantitative finance and development glossary · Python resources and libraries · Research sources and limitations