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Mathematics LibreTexts

Chapter 5: Integration

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    • 4.8: Antiderivatives
      At this point, we have seen how to calculate derivatives of many functions and have been introduced to a variety of their applications. We now ask a question that turns this process around: Given a function f , how do we find a function with the derivative f and why would we be interested in such a function?
    • 4.8E: Antiderivative and Indefinite Integral Exercises
    • 5.0: Prelude to Integration
    • 5.1: Approximating Areas
      In this section, we develop techniques to approximate the area between a curve, defined by a function f(x), and the x-axis on a closed interval [a,b]. Like Archimedes, we first approximate the area under the curve using shapes of known area (namely, rectangles). By using smaller and smaller rectangles, we get closer and closer approximations to the area. Taking a limit allows us to calculate the exact area under the curve.
    • 5.1E: Approximating Areas (Exercises)
    • 5.2: The Definite Integral
      If f(x) is a function defined on an interval [a,b], the definite integral of f from a to b is given by \[∫^b_af(x)dx=\lim_{n→∞} \sum_{i=1}^nf(x^∗_i)Δx,\] provided the limit exists. If this limit exists, the function f(x) is said to be integrable on [a,b], or is an integrable function. The numbers a and b are called the limits of integration; specifically, a is the lower limit and b is the upper limit. The function f(x) is the integrand, and x is the variable of integration.
    • 5.3: The Fundamental Theorem of Calculus
      The Fundamental Theorem of Calculus gave us a method to evaluate integrals without using Riemann sums. The drawback of this method, though, is that we must be able to find an antiderivative, and this is not always easy.
    • 5.4: Integration Formulas and the Net Change Theorem
      The net change theorem states that when a quantity changes, the final value equals the initial value plus the integral of the rate of change. Net change can be a positive number, a negative number, or zero. The area under an even function over a symmetric interval can be calculated by doubling the area over the positive x-axis. For an odd function, the integral over a symmetric interval equals zero, because half the area is negative.
    • 5.5: Substitution
      In this section we examine a technique, called integration by substitution, to help us find antiderivatives. Specifically, this method helps us find antiderivatives when the integrand is the result of a chain-rule derivative.
    • 5.5E and 5.6E U-Substitution Exercises
      These are homework exercises to accompany Chapter 5 of OpenStax's "Calculus" Textmap.
    • 5.6: Integrals Involving Exponential and Logarithmic Functions
      Exponential and logarithmic functions arise in many real-world applications, especially those involving growth and decay. Substitution is often used to evaluate integrals involving exponential functions or logarithms.
    • 5.7: Integrals Resulting in Inverse Trigonometric Functions and Related Integration Techniques
      Recall that trigonometric functions are not one-to-one unless the domains are restricted. When working with inverses of trigonometric functions, we always need to be careful to take these restrictions into account. Also in Derivatives, we developed formulas for derivatives of inverse trigonometric functions. The formulas developed there give rise directly to integration formulas involving inverse trigonometric functions.
    • 5.7E: Exercises for Integrals Resulting in Inverse Trigonometric Functions
    • 5.E: Integration (Exercises)
      These are homework exercises to accompany OpenStax's "Calculus" Textmap.