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Chapter 5: Integration

Last updated

Dec 21, 2020
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- 4.8E: AntiDerivative & Indefinite Integral Exercises
- 5.0: Prelude to Integration

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Topic hierarchy

5.0: Prelude to Integration
5.0E: Exercises
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].
5.1 Approximating Area (Riemann Sum) 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.
5.2 E: Definite Integral Intro Exercises
5.2 Exercises - these are good now
5.3: The Fundamental Theorem of Calculus Basics
5.3 E: FTOC Exercises
5.4: Average Value of a Function
$f_{ave}=\dfrac{1}{b−a}∫^b_af(x)dx.$
5.4E: Average Value of a Function Exercises
5.5: U-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 & 5.6E U-Substitution Exercises
These are homework exercises to accompany Chapter 5 of OpenStax's "Calculus" Textmap.
5.6: More U-Substitution - Exponential and Logarithmic Functions
Also U-Substitution for Exponential and logarithmic functions.
5.6 Notes
Notes on Chapter 5 Exercises.
5.7: Net Change
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.
5.7E: Net Change Exercises
These are homework exercises to accompany Chapter 5 of OpenStax's "Calculus" Textmap.
Xtra full 5.3: includes Proof of 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.

Contributors

Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org.

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