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1: Integration

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    Calculus is built on two operations — differentiation and integration.

    • Differentiation — as we saw last term, differentiation allows us to compute and study the instantaneous rate of change of quantities. At its most basic it allows us to compute tangent lines and velocities, but it also led us to quite sophisticated applications including approximation of functions through Taylor polynomials and optimisation of quantities by studying critical and singular points.
    • Integration — at its most basic, allows us to analyse the area under a curve. Of course, its application and importance extend far beyond areas and it plays a central role in solving differential equations.

    It is not immediately obvious that these two topics are related to each other. However, as we shall see, they are indeed intimately linked.

    • 1.1: Definition of the Integral
      Arguably the easiest way to introduce integration is by considering the area between the graph of a given function and the \(x\)-axis, between two specific vertical lines — such as is shown in the figure above. We'll follow this route by starting with a motivating example.
    • 1.2: Basic properties of the definite integral
      When we studied limits and derivatives, we developed methods for taking limits or derivatives of “complicated functions” like \(f(x)=x^2 + \sin(x)\) by understanding how limits and derivatives interact with basic arithmetic operations like addition and subtraction.
    • 1.3: The Fundamental Theorem of Calculus
      We have spent quite a few pages (and lectures) talking about definite integrals, what they are (Definition 1.1.9), when they exist (Theorem 1.1.10), how to compute some special cases (Section 1.1.5), some ways to manipulate them (Theorem 1.2.1 and 1.2.3) and how to bound them (Theorem 1.2.13).
    • 1.4: Substitution
      In the previous section we explored the fundamental theorem of calculus and the link it provides between definite integrals and antiderivatives. Indeed, integrals with simple integrands are usually evaluated via this link.
    • 1.5: Area between curves
      Before we continue our exploration of different methods for integrating functions, we have now have sufficient tools to examine some simple applications of definite integrals.
    • 1.6: Volumes
      Another simple application of integration is computing volumes. We use the same strategy as we used to express areas of regions in two dimensions as integrals — approximate the region by a union of small, simple pieces whose volume we can compute and then take the limit as the “piece size” tends to zero.
    • 1.7: Integration by parts
      The fundamental theorem of calculus tells us that it is very easy to integrate a derivative. In particular, we know that
    • 1.8: Trigonometric Integrals
      Integrals of polynomials of the trigonometric functions \(\sin x\text{,}\) \(\cos x\text{,}\) \(\tan x\) and so on, are generally evaluated by using a combination of simple substitutions and trigonometric identities.
    • 1.9: Trigonometric Substitution
      In this section we discuss substitutions that simplify integrals containing square roots of the form
    • 1.10: Partial Fractions
      Partial fractions is the name given to a technique of integration that may be used to integrate any rational function. We already know how to integrate some simple rational functions
    • 1.11: Numerical Integration
      In this section we turn to the problem of how to find (approximate) numerical values for integrals, without having to evaluate them algebraically. To develop these methods we return to Riemann sums and our geometric interpretation of the definite integral as the signed area.
    • 1.12: Improper Integrals
      To this point we have only considered nicely behaved integrals \(\int_a^b f(x)\, d{x}\text{.}\) Though the algebra involved in some of our examples was quite difficult, all the integrals had
    • 1.13: More Integration Examples
      Recall that we are using \(\log x\) to denote the logarithm of \(x\) with base \(e\text{.}\) In other courses it is often denoted \(\ln x\text{.}\)

    This page titled 1: Integration is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Joel Feldman, Andrew Rechnitzer and Elyse Yeager via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

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