Solution a copyright c 2011 the university of sydney

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Unformatted text preview: ght c 2011 The University of Sydney 1 5. The double integral of the function = x2 + 2y over the region of the xy plane in the first quadrant, bounded by the curves y = 1 - x2 , y = 0 and x = 0, is given by: 1 1-x2 (a) 0 1 0 1 x2 + 2y dx dy x2 + 2y dy dx 0 1 1-x2 0 1 1 1-y 1-y (b) (c) 0 x2 + 2y dx dy x2 + 2y dy dx. 0 (d) Solution: (c) 6. Which of the following integrals gives the length of the curve r t = cos t i + sin t j + t2 k, 1 2 t : 0 2 ? (a) 0 2 1 + 4t2 dt 1 + 4t 0 2 (b) 0 1 1 + 4t2 dt (c) Solution: (b) dt (d) 0 1 + 4t2 dt. =/2 =/2 7. Evaluate =0 =0 cos3 sin dd. (b) 8 (c) - 6 (d) - . 8 (a) 6 Solution: (b) 8. Let F = (2ax + y) i + (2y + z) j + (x - 6az) k, where a is a real constant. Find any values of a for which F = 0. 1 1 (a) a = (b) a = 2 4 (c) F = 0 for all values of a. (d) F is not zero for any a. Solution: (a) 2 9. Calculate (x2 y i + 2yz j + xz 2 k). (a) -2y i - z 2 j - x2 k (c) -2y i + z 2 j - x2 k (d) -2y i - z 2 j...
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