Calculus Volume 2

Key Terms

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
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