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# vctut04 - 5 Evaluate the following surface integral ii S x...

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The University of Sydney School of Mathematics and Statistics Tutorial 4 MATH2061: Vector Calculus Summer School 2012 1. Find ∇ · F in each of the following: (a) F = 2 i + 3 j + 4 k (b) F = x 2 yz i + xy 2 z j + xyz 2 k 2. Calculate the flux of F = 2 xy i + xy 3 j outwards across the curve C shown in the diagram. The curved part of C is an arc of the circle x 2 + y 2 = 4. The straight lines are y = x and y = x . y x C 3. Find the surface area of the part of the paraboloid z = 10 x 2 y 2 which lies above the xy -plane. 4. Let S be the surface defined by the following set of points in R 3 : { ( x,y,z ) | x + z = 5 , 0 x 3 , 0 y 4 } . (a) Sketch S , and its projection onto the xy -plane. (b) Find the cost of painting this surface if the cost is \$( xy + z 2 ) per unit area. (Cost = integraldisplayintegraldisplay S ( xy + z 2 ) dS .) 5. Evaluate the following surface integral:

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Unformatted text preview: 5. Evaluate the following surface integral: ii S ( x + z ) dS, S : x 2 + y 2 = 1 , ≤ z ≤ 1 . Copyright c c 2012 The University of Sydney 1 Extra questions for outside the tutorial 6. Find ∇ · F if F = x i ( x 2 + y 2 ) 3 / 2 + y j ( x 2 + y 2 ) 3 / 2 . 7. Find the surface area of the portion of the plane x a + y b + z c = 1 cut o± by x = 0 , y = 0 and z = 0 , where a > , b > , c > . 8. Find the surface area of the portion of the surface z = 2 3 ( x 3 / 2 + y 3 / 2 ) that lies above the triangle enclosed by the lines x = 0 , y = 0 and 2 x + 3 y = 6 . 2...
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vctut04 - 5 Evaluate the following surface integral ii S x...

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