mat293_q1_2006_solutions - J ~ A J(I 2~ 2!A x xL 3 Given F...

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MAT293F VECTOR CALCULUS Quiz 1 2 October 2006 9: 10 am - 10:OO am Closed Book, No aid sheets, No calculators Instructor: J. W. Davis Last Name: Given Name: SO\ & C M S Student #: FOR MARKER USE ONLY 11 TOTAL 11 Note: The following integrals may be useful. Page 1 of 7
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1) a) (4 marks) (5 marks) 1 Use polar coordinates to evaluate dA , where S is the 1" quadrant S section of the circIe 2 + J? = 4 between y = 0 and y = x. Sketch the region of integration. Evaluate [[La where R is the region given by: Osxs 1, 1 sys2. R Y Page 2 of 7
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2) Evaluate ly2 dV ,where V is the volume in the first octant bounded by the cylinders v 2 + y = 1 and z2 + y = 1. Make a sketch of the volume. Hint: the integral is much easier to solve if it is set up so that you integrate with respect toy last. (6 marks) Page 3 of 7
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Unformatted text preview: J , ~ A J (I - 2~ +- 2) !A x xL 3) Given F ( x ) = (-- + eY)& , find dFldx in two ways: a) by differentiating fust and b) by Y integrating first. ( .? marks) Page 4 of 7 4) Use a triple integral in spherical coordinates to find the volume of the solid common to the spheres p = 2&cos( and p = 2 . (1 0 marks) Page 5 of 7 L 5 ) Find the surface area of the part of the saddle az = 2 - 3 inside the cylinder 2 + J? = a2, a > 0, by means of a double integral over the projected area. . lice, . (6 marks) .*. Try Page 6 of 7 6) Let F '(x) = f(x) and G '(x) = g(x) on the interval asxs b. Let T be the triangle with vertices (I, a), (b, a) and (b, b). By integrating 11 f (x)&)d~ in both orders, show that: T (This is an alternate derivation of the formula for integration by parts.) (10 marks) Page 7 of 7...
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