# 9 the constrained extrema of fxy subject to the

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9. The constrained extrema of f(x,y) subject to the constraint g(x,y) = 0 will occur at a point where f (x 0 , y 0 ) = λ . g(x 0 , y 0 ) provided g(x 0 , y 0 ) is not 0 . We must also check the endpoints of g(x,y) = 0; and at the points where g= 0 or f = 0 , if these lie on g(x,y) = 0. 11. ∫∫ R f(x,y) dA = [a,b] { [g(x), h(x)] f(x,y).dy } dx = [c,d] { [p(y), q(y)] f(x,y).dx } dy ∫∫ R f(x,y) dA = [α,β] { [g(θ), h(θ)] f(r,θ) . r.dr } 12. Volume of the solid determined by z 1 (x,y) & z 2 (x,y) over the plane region R is given by Volume(G) = ∫∫ R { z 2 (x,y) - z 1 (x,y) } .dA . Area (R) = ∫∫ R 1 .dA . 13. Surface Area of the surface z = f(x,y) is given by: S = ∫∫ R {(f x ) 2 + (f y ) 2 + 1}. dA Surface Area of the surface r = x(u,v), y(u,v), z(u,v) is: ∫∫ R ||( r / u)×( r / v)|| . dA 14. Mass of the solid region G with density f(x,y,z) at x, y, z is given by. Mass(G) = ∫∫∫ G f(x,y,z) .dV . Volume (G) = ∫∫∫ G 1 .dV .
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