# Integral in cartesian coordinates that would be used

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integral in cartesian coordinates that would be used to find the volume of the solid G bounded below by the elliptic paraboloid z = 4 x 2 + y 2 and above by the cylindrical surface z = 4 - 3 y 2 , [Hint: The upper and lower surfaces are on a platter with a cherry on top. Find the projection onto the xy-plane of the curve obtained when the two surfaces intersect.]

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TEST3/MAC2313 Page 3 of 5 ______________________________________________________________________ 5. (10 pts.) Write down the triple iterated integral in cylindrical coordinates that would be used to compute the volume of the solid G whose top is the plane z = 9 and whose bottom is the paraboloid z = x 2 + y 2 . Do not attempt to evaluate the integral. ______________________________________________________________________ 6. (10 pts.) Locate all relative extrema and saddle points of the following function. f ( x , y ) x 2 8 y 2 x 2 y Use the second partials test in making your classification. (Fill in the table below after you locate all the critical points.) Crit.Pt. f xx @ c.p. f yy @ c.p. f xy @ c.p. D @ c.p. Conclusion
TEST3/MAC2313 Page 4 of 5 ______________________________________________________________________ 7. (5 pts.) Write down the triple iterated integral in spherical coordinates that would be used to compute the volume of the solid G bounded above by the sphere defined by ρ = 4 and below by the cone defined by φ = π /3. Do not attempt to evaluate the interated integrals. ______________________________________________________________________ 8. (15 pts.) Let f ( x , y ) = x 2 y 2 on the closed unit disk defined by x 2 + y 2 1. Find the absolute extrema and where they occur.

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