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# MATH 241 Exam 3 Study Guide (all concepts) Math 241

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This 8 page Study Guide was uploaded by Jigisha Sampat on Saturday April 16, 2016. The Study Guide belongs to Math 241 at University of Illinois at Urbana-Champaign taught by Dr. Jason Anema in Spring 2016. Since its upload, it has received 192 views. For similar materials see Calculus 3 in Mathematics (M) at University of Illinois at Urbana-Champaign.

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Date Created: 04/16/16

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 ???? 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 1 Fubini’s Theorem If ???? is continuous on the rectangle , then More generally, this is true if we assume that ???? is bounded on R, ???? 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 ???? formed by two curves. ???? = {(????,???? |???? ≤ ???? ≤ ????,???? ???? 1 ???? ≤ ???? ???? 2( )} ???? ???? = ????2???? )???? ????,???? ???????? ∫????1(????) ???? ???? = ∫???? ???? ???? ∙ ???????? ???? ????2(????) ( ) = ∫????∫ ????1(????)???? ????,???? ???????? ???????? 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 ???? ???? ) 2 ???? ???? = ∫ ???? ????,???? ∙???????? ????1???? ) ???? ???? = ∫????(????) ∙ ???????? ???? ???? ???? 2( ) = ∫???? ???? 1( )???? (????,???? )???????? ???????? 2 TYPE 3 DOMAIN: The entire figure is a closed curve with no fixed end points: Steps to Solve: 1) Pick an axis and find the extreme points of the closed figure 2) Draw a line through any one of these points, dividing the region D in to D1 and D2 3) Both D1 and D2 can be solved as TYPE 2 domains 4) Volume of the total domain D = D1 + D2, provided that g1 D1 ∪ D2 = D (there is union of the entire region D1 ∩ D2 = null (Ф) (no overlap) Properties of Double Integrals: 3 Comparison of Co-ordinate Systems Type Polar Co-ordinates Spherical Co-ordinates Cylindrical Co-ordinates Used for Double Integral Triple Integral Triple Integral Format ???? = ???? cos???? ???? = ????sin∅ ???????????????? ???? = ???? cos???? ???? = ????sin???? ???? = ????sin∅sin???? ???? = ????sin???? ???? + ???? + ???? = ????2 2 ???? = ????cos∅ ???? = ???? Graph Differential ???????? = ???? ???????? ????θ ???????? = ???? ???????????? ∅ ???????? ????θ d∅ ???????? = ???? ???????? ???????? ???????? Used to surface areas of circular figurevolumes enclosed by curves, Volumes enclosed by curves simplify and volume under a curve especially regions bounded by especially when they can be calculations spheres and cones simplifies to polar coordinates of in one plane Double Integration: Polar v/s variable change Double Integration with polar Double Integration with polar co-ordinates Type co-ordinates Used Simple surfaces with fixed Surfaces bounded by curves with no defined limits to limits in one coordinate Solve Graphs Steps 1) Sketch region bounded 1) Transform region S into T by expressing ???? and ???? as functions of ???? to by curves defined in and ???? Solve the question 2) Pick the functions for ???? and ???? by solving the equations for the 2) Identify limits for ???? and boundaries of the given region simultaneously (make sure that ???? ???? and ???? are easily replaceable in ????(????,????) 3) Substitute ???? = 3) Evaluate the limits for ???? and ???? ????cos???? 4) Calculate the Jacobian (see below) ???? = ????sin???? 5) Replace ???????? by the Jacobian times ???????? ???????? 4) Simplify ????(????,????) using 6) Evaluate the integral of ????(????,????) above substitution 5) Replace ???????? by ???? ???????? ???????? 6) Evaluate the integral 4 Calculating the Jacobian Applications of Double Integrals Density and mass Mass Moments and Centers of Mass Moment around ???? Moment around ???? ???? co-ordinate of Center of Mass ???? co-ordinate of Center of Mass Moment of inertia Moment of Inertia about ????- axis Moment of Inertia about ????- axis Moment of Inertia about the origin 5 Triple Integrals over Spherical and Cylindrical coordinates Type Cylindrical Spherical Steps 1) Sketch graph of equation 1) Sketch the graph to 2) Identify limits for ???? 2) Identify ???? as√???? + ???? + ???? and find limits of ???? Solve 3) Substitute ???? = ????cos???? 3) Substitute ???? = ????sin∅ ???????????????? ???? = ????sin???? ???? = ????sin∅sin???? 4) Find limits for ???? and ???? ???? = ????cos∅ 5) Express ????(????,????,????) as 4) Find limits of ∅ and ???? by simultaneously solving ????(????cos????,????sin????,????) above equations 6) Write ???????? as ???? ???????? ???????? ???????? 5) Express ????(????,????,????) as 7) Evaluate triple integral ????(????sin∅ ????????????????,????sin∅sin????,????cos∅) 6) Write ???????? as ???? sin∅ ???????? ???????? ????∅ 7) Evaluate the Triple Integral Application of Triple Integrals Mass Moment Moment around ???????? plane Moment around ???????? plane Moment around ???????? plane Center of Mass ???? co-ordinate of Center of Mass ???? co-ordinate of Center of Mass ???? co-ordinate of Center of Mass If the density is constant, the center of mass is called the Centroid. 6 Line Integrals Fundamental Theorem Green’s Theorem Used when Only on simple surfaces that are closed Can be applied to simple surfaces as well as continuous (i.e. Do not intersect, or cosurfaces with holes holes) Theorem Let ???? be a smooth curve givn by the VectLet ???? be a positively oriented, piece-wise function ????(????), where ???? ≤ ???? ≤ ????. Let ???? besmooth, simple curve in the plane and let ???? differentiable function of two or three be the region bounded by ????. If ???? and ???? variables whose gradient vector ∇????, is have continuous partial derivatives on an continuous on ????. Then, open region that contains ????, then ∫ ∇???? ∙ ???????? = ????(???? ????))− ????(???? ???? ) ∫ ???? ???????? + ???? ???????? = ∬( ???????? − ???????? )???????? ???? ???? ???????? ???????? ???? Path It is path independent as long as The pat chosen is along the boundary of the dependency ???? ∙ ???????? = 0 for every closed path ???? region and the direction of path matters in ∫???? determining sign of the area How to Identify Conservative Field 1. If ???? is a continuous vector field on an open connected reg∫????n???? ∙ ???????? path independent in ????, then ???? is a conservative vector field on D. And there exists a function such that ∇???? = ????. OR 2. Let ???? = ???? ???? + ???? ???? be a vector field on an open simply-connected region ????. Suppose that ???? and ???? have continuous first-order partial derivatives and ???????? ???????? = ???????? ???????? Throughout ????. Then ???? is conservative. Steps to apply Green’s Theorem Green’s Theorem Extended 1. Sketch the graph and identify its orientation (clockwise or anti-clockwise) 2. Identify ???????? and ???????? Identify ???? ????,???? and ???? ????,???? ???????? ???????? 3. Calculate and ???????? ???????? 4. Find limits of ???? and ???? ???????? ???????? 5. Calculate∬ ???? (???????? − ???????? )???????? 6. Evaluate the integral and assign positive sign to your answer for anticlockwise orientation and negative sign for clockwise orientation 7

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