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UIUC - MATH 241 - Study Guide - Midterm

Created by: Jigisha Sampat Elite Notetaker

UIUC - MATH 241 - Study Guide - Midterm

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background image   MATH 241 EXAM 3 STUDY  GUIDE   By Jigisha Sampat and Kevin Thayyil      Contents  Double Integral of F .................................................................................................................................. 1  The Midpoint Rule..................................................................................................................................... 1  Fubini’s Theorem ...................................................................................................................................... 2  TYPE 1 DOMAIN: ................................................................................................................................... 2  TYPE 2 DOMAIN: ................................................................................................................................... 2  TYPE 3 DOMAIN: ................................................................................................................................... 3  Properties of Double Integrals: ................................................................................................................. 3  Comparison of Co-ordinate Systems ........................................................................................................ 4  Double Integration: Polar v/s variable change ......................................................................................... 4  Calculating the Jacobian........................................................................................................................ 5  Applications of Double Integrals ............................................................................................................... 5  Density and mass .................................................................................................................................. 5  Moments and Centers of Mass ............................................................................................................. 5  Moment of inertia ................................................................................................................................. 5  Triple Integrals over Spherical and Cylindrical coordinates ...................................................................... 6  Application of Triple Integrals ................................................................................................................... 6  Moment .................................................................................................................................................... 6  Center of Mass .......................................................................................................................................... 6  Line Integrals ............................................................................................................................................. 7  How to Identify Conservative Field ........................................................................................................... 7  Steps to apply Green’s Theorem ............................................................................................................... 7  Green’s Theorem Extended ...................................................................................................................... 7          
background image     Double Integral of F          
If this limit exists 
        1.  Represents the sum of volumes of  columns  2.  Is an approximation to the volume under the graph of 𝑓    
  
         The Midpoint Rule                
Steps to Estimating using Midpoint Rule: 
  
1.  Identify m and n, where m is the number of parts of x and n is the number of parts of y
2.  Set the limits of Riemann Sums as follows: 
  3.    4.    5.    6.  Substitute in the Midpoint Formula       
background image   Fubini’s Theorem   If 𝑓 is continuous on the rectangle  , then        More generally, this is true if we assume that 𝑓 is bounded on R, 𝑓 is discontinuous only 
on a finite number of smooth curves, and the iterated integral don’t exist.  
  Double integral calculations can be simplified by identifying one of the following three domain types:  TYPE 1 DOMAIN:    Where D is the area enclosed by a continuous function 𝑓 formed by two curves.    𝐷 = {(𝑥, 𝑦)|𝑎 ≤ 𝑥 ≤ 𝑏, 𝑔 1 (𝑥) ≤ 𝑦 ≤ 𝑔 2 (𝑥)}    𝐴(𝑥) =   ∫ 𝑓(𝑥, 𝑦)𝑑𝑦 𝑔 2 (𝑥) 𝑔 1 (𝑥)     𝑉 =   ∫ 𝐴(𝑥) ∙ 𝑑𝑥 𝑏 𝑎     = ∫ ∫ 𝑓(𝑥, 𝑦)𝑑𝑦 𝑑𝑥 𝑔 2 (𝑥) 𝑔 1 (𝑥) 𝑏 𝑎   TYPE 2 DOMAIN:    Where two curves intersect at 2 points:  Steps to Solve:  1)  Find the point of intersection and pick an axis with different  values for the two points (if possible)  2)  Continue as usual, according to axis     𝐴(𝑥) =   ∫ 𝑓 ( 𝑥, 𝑦 ) ∙ 𝑑𝑦 𝑔 2( 𝑥 ) 𝑔 1( 𝑥 )     𝑉 =   ∫ 𝐴(𝑥) ∙ 𝑑𝑥 𝑏 𝑎           =   ∫ ∫ 𝑓 ( 𝑥, 𝑦 ) 𝑑𝑦 𝑑𝑥 𝑔 2 ( 𝑥 ) 𝑔 1 ( 𝑥 ) 𝑏 𝑎      

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School: University of Illinois at Urbana - Champaign
Department: Math
Course: Calculus III
Professor: Professor Oikhberg
Term: Summer 2015
Tags: Green's, theorem, 241, theorem, midpoint, midpoint, fubini, fubini, fubini's theorem, fubini's theorem, triple integrals, triple integrals, Double Integrals, Double Integrals, line, line, Integrals, Integrals, polar coordinates, polar coordinates, cylindrical coordinates, cylindrical coordinates, spherical coordinates, spherical coordinates, domain, domain, domains, domains, applications, and applications
Name: MATH 241 Exam 3 Study Guide (all concepts)
Description: Every Concept you need to know for Exam 3, MATH 241, Spring 2016. Please try a sample problem under each concept, from webassign, class notes or Stewart. Click on a topic in the contents page to go directly to it!
Uploaded: 04/16/2016
8 Pages 146 Views 116 Unlocks
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