 Limited time offer 20% OFF StudySoup Subscription details

# ISU - MATH 166 - Math 166 Calculus 2 - Study Guide

### Created by: Rebecca Wilson Elite Notetaker

> > > > ISU - MATH 166 - Math 166 Calculus 2 - Study Guide

ISU - MATH 166 - Math 166 Calculus 2 - Study Guide

##### Description: This study guide covers what will be on the next exam. They cover sections 8.6, 8.7, and 10.1.
This preview shows pages 1 - 2 of a 4 page document. to view the rest of the content Math 166 - Calculus 2   Chapter 8: Techniques of Integration  8.6 | Numerical Integration
The Trapezoidal Rule:
To approximate   use (x)dx b a f   T =  (y y y .. y ) 2 Δx 0 + 2 1 + 2 2 + . + 2 n−1 y n   The y’s are the values of   f   ​at the partition points  , where x x, Δx, ... , x   n xx x 0 a 1 1 + Δ x 2 + 2     n−1 + ( − 1   n b       = (b-a)/n x Δ
Simpson’s Rule:
To approximate   use (x)dx b a f   S =  (y 4 y .. y y ) 3 Δx 0 + 2 + 2 2 + . + 2 n−2 + 4 n−1 y n   The y’s are the values of   f   ​at the partition points  , the number  n x x, Δx, ... , x   n xx x 0 a 1 1 + Δ x 2 + 2     n−1 + ( − 1   n b     is even, and   = (b-a)/n x Δ
Error Estimations:
If
f’’  ​is continuous and M is any upper bound for the values of   on [a,b], then the error in the f| ′′ |   approximation of the integral of  f ​ from ​a​ to ​b​ for n steps satisfies the inequality:   Trapezoidal Error:   E | | T | | ≤ 12n 2 M(ba) 3     Simpson’s Error:  E | | S | | ≤ 180n 4 M(ba) 4
8.7 | Improper Integrals
Integrals with infinite limits of integration are improper integrals of Type 1.
1. If f(x) is continuous on [a,  ), then   (xdx (xdx a f = lim b→∞ b a f   2. If f(x) is continuous on (- , b], then    (xdx (xdx b −∞ f = lim a→−∞ b a f 3. If f(x) is continuous on (- ,  ), then ∞ ∞    , where  c ​ is any number (xdx (xdx  (xdx −∞ f = lim a→∞ c a f +   lim b→∞ b c f   In each case, if the limit is finite we say that the improper integral  converges ​and that the limit is  the  value​ of the improper integral. If the limit fails to exist, the improper integral ​diverges​.
Integrals of functions that become infinite at a point within the interval of integration are
improper integrals of Type 2.
1. If f(x) is continuous on (a, b] and discontinuous at  a ​, then  (xdx (xdx b a f = lim ca + b c f   2. If f(x) is continuous on [a, b) and discontinuous at  b ​, then   (xdx (xdx b a f = lim cb c a f   3. If f(x) is discontinuous at c, where a < c < b, and continuous on [a,c) (c, b], then    (xdx (xdx  (xdx b a f = c a f +   b c f   In each case, if the limit is finite we say that the improper integral  converges ​and that the limit is  the  value​ of the improper integral. If the limit does not exist, the integral ​diverges​.
Direct Comparison Test:
Let
​and ​g​ be continuous on [a,  ) with 0  f(x)  g(x) for all x  a. Then,   1.  converges if   converges. (xdx a f (xdx a g   2.  diverges if   diverges. (xdx a g (xdx a f

Limit Comparison Test
If the positive functions f and g are continuous on [a,  ), and if
,  0 < L <  , then lim x→∞ f(x) g(x) L     and  both converge or both diverge. (xdx a f (xdx  a g          Chapter 10: Infinite Sequences and Series

This is the end of the preview. Please to view the rest of the content Join more than 18,000+ college students at Iowa State University who use StudySoup to get ahead
4 Pages 33 Views 26 Unlocks
• Better Grades Guarantee
• 24/7 Homework help
• Notes, Study Guides, Flashcards + More! Join more than 18,000+ college students at Iowa State University who use StudySoup to get ahead
##### Description: This study guide covers what will be on the next exam. They cover sections 8.6, 8.7, and 10.1.
4 Pages 33 Views 26 Unlocks
• Better Grades Guarantee
• 24/7 Homework help
• Notes, Study Guides, Flashcards + More!
Get Full Access to ISU - Math 166 - Study Guide - Midterm
×
Get Full Access to ISU - Math 166 - Study Guide - Midterm

I don't want to reset my password

Need help? Contact support

Need an Account? Is not associated with an account
We're here to help

Having trouble accessing your account? Let us help you, contact support at +1(510) 944-1054 or support@studysoup.com

Password Reset Request Sent An email has been sent to the email address associated to your account. Follow the link in the email to reset your password. If you're having trouble finding our email please check your spam folder
Already have an Account? Is already in use