Assignment 7

# Assignment 7 - f x,y = 4 x 2-3 y 2 2 xy show that f x,y...

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Math222 Assignment 7 due Friday Nov. 23, 2007 1. Compute double integral R R D xdA where D is the finite region bounded by y = 2 x 2 and y = 1 + x 2 . 2. Compute double integral R R D ( x 2 + y 2 ) dA where D is the finite region between y = x and y = x 2 . Integrate with respect to x first. 3. By reversing the order of integration evaluate Z 9 0 Z 3 y sin πx 3 dxdy. 4. Evaluate R 2 - 2 R 4 - x 2 - 4 - x 2 R 1 ( x 2 + y 2 ) 2 x 2 dzdydx . 5. Evaluate using polar coordinates: R D ydA , if D is the region in the first quadrant bounded by the circle x 2 + y 2 = 9 and the lines y = 0 and y = x . 6. Let Ω be the solid region bounded above by the plane y + z = 4 , below by the xy plane and on the sides by the cylinder x 2 + y 2 = 16. Evaluate Z Ω p x 2 + y 2 dV . 7. Let
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Unformatted text preview: f ( x,y ) = 4 x 2-3 y 2 + 2 xy , show that f ( x,y ) does not have a local max or local min anywhere in the plane. Does it have a saddle point? (Justify your answer.) Find the max and min of f ( x,y ) on the square { ( x,y ) | ≤ x ≤ 1 , ≤ y ≤ 1 } , naming the points at which these extrema occur. 8. Use the method of Lagrange multipliers (or otherwise) to ﬁnd maxima and minima of f ( x,y,z ) = x 2 + y 2 + z 2 on the ellipse formed by intersection of the cone z 2 = x 2 + y 2 by the plane x-2 z = 3....
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