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Calculus I / The Integral

Overview

Differentiation asks: given a function, how fast is it changing? Integration asks the inverse question: given a rate of change, how much has accumulated? A continuously compounded return rate r(t)r(t) accumulates total log-return ∫0Tr(t) dt\int_0^T r(t)\,dt; a stream of cash flows C(t)C(t) accumulates to total present value ∫0TC(t)e−rt dt\int_0^T C(t)e^{-rt}\,dt. The definite integral is the mathematical object that makes these statements precise.
The central result of this topic is the Fundamental Theorem of Calculus (FTC), which links differentiation and integration: computing a definite integral reduces to evaluating an antiderivative at two points. This makes the entire antidifferentiation toolkit from Topic 4 into the engine for computing signed areas, total changes, and accumulated quantities.

By the end of this topic you should be able to:

  • form Riemann sums and take the limit to define the definite integral;
  • state both parts of the FTC and use Part II to evaluate definite integrals;
  • apply the Net Change Theorem and uu-substitution to a broad class of integrals;
  • interpret definite integrals as accumulated P&L, total return, or expected value.

In this lesson

  1. Contents
  2. The big picture

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