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# UIUC - MATH 241 - Study Guide - Midterm

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UIUC - MATH 241 - Study Guide - Midterm

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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
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
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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##### 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!
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