StudyGuide2Updated - MA 261 - Fall 2009 Study Guide # 2...

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Unformatted text preview: MA 261 - Fall 2009 Study Guide # 2 (updated) 1. Relative/local extrema; critical points ( f = ~ or f does not exist); 2 nd Derivatives Test; absolute extrema; Max-Min Problems; Lagrange Multipliers: Extremize f ( ~ x ) subject to a constraint g ( ~ x ) = C , solve the system: f = g and g ( ~ x ) = C . 2. Double integrals; Midpoint Rule for rectangle : Z Z R f ( x, y ) dA m X i =1 n X j =1 f ( x i , y j ) A ; 3. Type I region D : g 1 ( x ) y g 2 ( x ) a x b ; Type II region D : h 1 ( y ) x h 2 ( y ) c y d ; iterated integrals over Type I and II regions: Z Z D f ( x, y ) dA = Z b a Z g 2 ( x ) g 1 ( x ) f ( x, y ) dy dx and ZZ D f ( x, y ) dA = Z d c Z h 2 ( y ) h 1 ( y ) f ( x, y ) dx dy , respectively; Reversing Order of Integration (regions that are both Type I and Type II); properties of double integrals. 4. Integral inequalities: mA Z Z D f ( x, y ) dA MA , where A = area of D and m f ( x, y ) M on D . 5. Change of Variables Formula in Polar Coordinates: if D : h 1 ( ) r h 2 ( ) , then ZZ D f ( x, y ) dA = Z Z h 2 ( ) h 1 ( ) f ( r cos , r sin ) r dr d . 6. Applications of double integrals: (a) Area of region D is A ( D ) = ZZ D f ( x, y ) dA (b) Volume of solid under graph of z = f ( x, y ), where f ( x, y ) 0, is V = Z Z D f ( x, y ) dA (c) Mass of D is m = Z Z D ( x, y ) dA , where ( x, y ) = density (per unit area); sometimes write m = ZZ D dm , where dm = ( x, y ) dA . (d) Moment about the x-axis M x = ZZ D y ( x, y ) dA ; moment about the y-axis M y = ZZ D x ( x, y ) dA . (e) Center of mass ( x, y ), where x = M y m = ZZ D x ( x, y ) dA ZZ D ( x, y ) dA , y = M x m = ZZ D y ( x, y ) dA Z Z D ( x, y ) dA Remark : centroid = center of mass when density is constant (this is useful)....
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This note was uploaded on 11/13/2011 for the course MA 261 taught by Professor Stefanov during the Fall '08 term at Purdue University-West Lafayette.

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StudyGuide2Updated - MA 261 - Fall 2009 Study Guide # 2...

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