Skip to Content
Calculus Volume 2

# Key Terms

absolute error
if $BB$ is an estimate of some quantity having an actual value of $A,A,$ then the absolute error is given by $|A−B||A−B|$
computer algebra system (CAS)
technology used to perform many mathematical tasks, including integration
improper integral
an integral over an infinite interval or an integral of a function containing an infinite discontinuity on the interval; an improper integral is defined in terms of a limit. The improper integral converges if this limit is a finite real number; otherwise, the improper integral diverges
integration by parts
a technique of integration that allows the exchange of one integral for another using the formula $∫​udv=uv−∫​vdu∫​udv=uv−∫​vdu$
integration table
a table that lists integration formulas
midpoint rule
a rule that uses a Riemann sum of the form $Mn=∑i=1nf(mi)Δx,Mn=∑i=1nf(mi)Δx,$ where $mimi$ is the midpoint of the ith subinterval to approximate $∫abf(x)dx∫abf(x)dx$
numerical integration
the variety of numerical methods used to estimate the value of a definite integral, including the midpoint rule, trapezoidal rule, and Simpson’s rule
partial fraction decomposition
a technique used to break down a rational function into the sum of simple rational functions
power reduction formula
a rule that allows an integral of a power of a trigonometric function to be exchanged for an integral involving a lower power
relative error
error as a percentage of the absolute value, given by $|A−BA|=|A−BA|·100%|A−BA|=|A−BA|·100%$
Simpson’s rule
a rule that approximates $∫abf(x)dx∫abf(x)dx$ using the integrals of a piecewise quadratic function. The approximation $SnSn$ to $∫abf(x)dx∫abf(x)dx$ is given by $Sn=Δx3(f(x0)+4f(x1)+2f(x2)+4f(x3)+2f(x4)+4f(x5)+⋯+2f(xn−2)+4f(xn−1)+f(xn))Sn=Δx3(f(x0)+4f(x1)+2f(x2)+4f(x3)+2f(x4)+4f(x5)+⋯+2f(xn−2)+4f(xn−1)+f(xn))$ trapezoidal rule a rule that approximates $∫abf(x)dx∫abf(x)dx$ using trapezoids
trigonometric integral
an integral involving powers and products of trigonometric functions
trigonometric substitution
an integration technique that converts an algebraic integral containing expressions of the form $a2−x2,a2−x2,$ $a2+x2,a2+x2,$ or $x2−a2x2−a2$ into a trigonometric integral
Citation/Attribution

Want to cite, share, or modify this book? This book is Creative Commons Attribution-NonCommercial-ShareAlike License 4.0 and you must attribute OpenStax.

Attribution information
• If you are redistributing all or part of this book in a print format, then you must include on every physical page the following attribution:
Access for free at https://openstax.org/books/calculus-volume-2/pages/1-introduction
• If you are redistributing all or part of this book in a digital format, then you must include on every digital page view the following attribution:
Access for free at https://openstax.org/books/calculus-volume-2/pages/1-introduction
Citation information

© Mar 30, 2016 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike License 4.0 license. The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.