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StudyGuide2Updated

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

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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 = λ ∇ 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 ) Δ 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 ( θ ) ≤ r ≤ h 2 ( θ ) α ≤ θ ≤ β , then ZZ D f ( x, y ) dA = Z β α Z h 2 ( θ ) h 1 ( θ ) f ( r cos θ, r sin θ ) r dr dθ . ↑ 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 ρ ( x, y ) dA , where ρ ( x, y ) = density (per unit area); sometimes write m = ZZ D dm , where dm = ρ ( x, y ) dA . (d) Moment about the x-axis M x = ZZ D y ρ ( x, y ) dA ; moment about the y-axis M y = ZZ D x ρ ( x, y ) dA . (e) Center of mass ( x, y ), where x = M y m = ZZ D x ρ ( x, y ) dA ZZ D ρ ( x, y ) dA , y = M x m = ZZ D y ρ ( x, y ) dA Z Z D ρ ( x, y ) dA Remark : centroid = center of mass when density is constant (this is useful)....
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StudyGuide2Updated - MA 261 Fall 2009 Study Guide 2(updated...

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