Tutorial 7

# Tutorial 7 - y 1 on the bottom by the xy-plane and on the...

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MATH1211/10-11(1))/Tu7/ THE UNIVERSITY OF HONG KONG DEPARTMENT OF MATHEMATICS MATH1211 Multivariable Calculus 2010-11 First Semester: Tutorial 7 1. Let f : S R 2 R be a scalar-valued function deﬁned on a subset S of R 2 . Prove or disprove the following statements. (a) If f is continuous and has a global maximum and a global minimum on S , then S is compact. (b) If f does not have a global minimum on S , then S is not compact. 2. Evaluate Z π 0 Z 2 1 y sin xdy dx . 3. ( § 5.1 no.8) Find the volume of the region bounded on top by the plane z = x + 3
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Unformatted text preview: y + 1, on the bottom by the xy-plane, and on the sides by the planes x = 0 , x = 3 , y = 1 , y = 2. 4. Suppose that f is a nonnegative-valued, continuous function deﬁned on R = { ( x,y ) | a ≤ x ≤ b, c ≤ y ≤ d } . If f ( x,y ) ≤ M for some positive number M , explain why the volume V under the graph of f over R is at most M ( b-a )( d-c ). 5. Integrate the function f ( x,y ) = 3 xy over the region bounded by y = 32 x 3 and y = √ x . 1...
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