MAC 2313 Review 3

# MAC 2313 Review 3 - 5) Sketch the region of integration and...

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MAC 2313 SUMMER 2010 Test-3 Review 1) Find the local maximum and minimum values and saddle points of the following functions. (i) f ( x,y ) = e 4 y - x 2 - y 2 . (ii) f ( x,y ) = 9 - 2 x + 4 y - x 2 - 4 y 2 . 2) Using the method of Lagrange multipliers, ﬁnd the maximum and minimum values of the following functions subject to the given constraints. (i) f ( x,y ) = 4 x + 6 y , x 2 + y 2 = 13. (ii) f ( x,y ) = 2 x 2 + 3 y 2 - 4 x - 5, x 2 + y 2 16. 3) Let f ( x,y ) be the function in Problem #2 (ii). Find upper and lower bounds for the dou- ble integral R R D f ( x,y ) dA where D = { ( x,y ) : x 2 + y 2 16 } without actually evaluating it. 4) Using double integrals, ﬁnd the volume of (i) the solid that lies under the surface z = xye x 2 y and above the rectangular region R = { ( x,y ) : 0 x 1 , 0 y 2 } . (ii) the solid in the ﬁrst octant bounded by the cylinder z = 16 - x 2 and the plane y = 5. (iii) the solid that lies under the surface z = 2 x + y 2 and above the region bounded by the curves x = y 2 and x = y 3 . (iv) the solid that is bounded by the cylinder x 2 + y 2 = 1 and the planes y = z , x = 0, z = 0 in the ﬁrst octant.

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Unformatted text preview: 5) Sketch the region of integration and change the order of integration for the integral R 3 R 9-y f ( x,y ) dxdy . 6) Using Polar Coordinates, evaluate R 2 R 2 x-x 2 p x 2 + y 2 dydx . 7) Let E denote the solid bounded by the surfaces y = 4-x 2-4 z 2 and y = 0. Express the integral R R R E f ( x,y,z ) dV by interpreting dV as (i) dydzdx (ii) dzdydx (iii) dxdydz 1 8) Evaluate (i) the volume of the solid bounded by the surfaces y = x 2 , z = 0, z = 4, and y = 9. (ii) Z 2-2 Z 4-y 2- 4-y 2 Z 2 x 2 + y 2 xz dzdxdy (iii) R R R E xyz dV where E lies between the spheres x 2 + y 2 + z 2 = 4 and x 2 + y 2 + z 2 = 16 and above the cone = 3 . (iv) Z a-a Z a 2-y 2- a 2-y 2 Z a 2-x 2-y 2- a 2-x 2-y 2 ( x 2 z + y 2 z + z 3 ) dzdxdy . 2...
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## This note was uploaded on 12/15/2011 for the course MAC 2313 taught by Professor Keeran during the Spring '08 term at University of Florida.

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MAC 2313 Review 3 - 5) Sketch the region of integration and...

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