quiz2A-2011 - The University of Sydney School of...

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The University of Sydney School of Mathematics and Statistics Solutions to Vector Calculus - Quiz A MATH2061/2067: Semester 1, 2011 1. Let F = 2 x i + y j . Find the fux c C F · n ds , where C is the unit circle, centre (0 , 0), taken once anti-clockwise, and n the unit outward normal. (a) 0 (b) π/ 2 (c) 3 π (d) 2 π . Solution: (c) 2. Which o± the ±ollowing integrals gives the volume o± the solid enclosed between the paraboloid z = 1 x 2 y 2 , z 0, and the xy -plane ? (a) i θ =2 π θ =0 i r =1 r =0 r 2 dr dθ (b) i θ =2 π θ =0 i r =1 r =0 (1 r 2 ) 1 / 2 r dr dθ (c) i θ =2 π θ =0 i r =1 r =0 (1 r 2 ) r dr dθ (d) i θ =2 π θ =0 i r =1 r =0 (1 r 2 ) 1 / 2 dr dθ . Solution: (c) 3. Find a potential ±unction o± the gradient ²eld F = ( y 2 (sin x ) e z ) i + 2 xy j + (cos x ) e z k . (a) xy 2 + (sin x ) e z (b) 2 xy + (sin x ) e z (cos x ) e z (c) 2 xy (sin x ) e z (d) xy 2 + (cos x ) e z . Solution: (d) 4. Evaluate ii R ( x + 2 y ) dA where R is the shaded region shown in the dia- gram. 1 1 x y (a) 1 / 2 (b) 4 / 3 (c) 3 / 2 (d) 2 / 3. Solution: (a) Copyright c c 2011 The University of Sydney 1
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5. The double integral of the function φ = x 2 + 2 y over the region of the xy plane in the Frst quadrant, bounded by the curves y = 1 x 2 , y = 0 and x = 0, is given by: (a) i 1 0 bi 1 x 2 0 ( x 2 + 2 y ) dx B dy (b) i 1 0 bi 1 1 x 2 ( x 2 + 2 y ) dy B dx (c) i 1 0 bi 1 y 0 ( x 2 + 2 y ) dx B dy (d) i 1 0 bi 1 y 1 ( x 2 + 2 y ) dy B dx .
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quiz2A-2011 - The University of Sydney School of...

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