5: Integration
- Page ID
- 32810
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- 5.2: 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].
- 5.4: 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.
- 5.5: Definite Integral Intro Exercises
- 5.2 Exercises - these are good now
- 5.8: Average Value of a Function
- \[f_{ave}=\dfrac{1}{b−a}∫^b_af(x)dx.\]
- 5.9: 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.
- 5.10: 5.5E and 5.6E U-Substitution Exercises
- These are homework exercises to accompany Chapter 5 of OpenStax's "Calculus" Textmap.
- 5.11: More U-Substitution - Exponential and Logarithmic Functions
- Also U-Substitution for Exponential and logarithmic functions.
- 5.12: 5.6 Notes
- Notes on Chapter 5 Exercises.
- 5.13: 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.
- 5.14: 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.
- 5.7E: Net Change Exercises
- These are homework exercises to accompany Chapter 5 of OpenStax's "Calculus" Textmap.
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.