METU NCC Mat 120 Spring 2010 Final - 5(6 6 6 pts For each of the vector&elds below check whether it is conservative or not Find a potential function

Last Name Name Student No Department: Section: Signature: : : : : : : : : : : Code Acad.Year Semester Date Time Duration M E T U Northern Cyprus Campus Calculus for Functions of Several Variables Final Math 120 2009-2010 Spring 4.6.2010 9:30 120 minutes 6 QUESTIONS ON 6 PAGES TOTAL 100 POINTS 1 2 3 4 5 6 1 (4+12 pts.). Let f ( x ) = 2 ° x and A the total shaded area below. (a) Express A as an in°nite series. (b) Does this series converge? If so, what is its value?

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2.(16 pts.) Find the shortest distance between the plane 4 x + 2 y ° z = 20 and the paraboloid z = x 2 + y 2 , using Lagrange Multipliers Method.

3.(16 pts.) Calculate the double integral Z Z R ° 12 y 2 ° 12 xy ° 24 x 2 ± dx dy over the parallelogram R bounded by the lines y ° 2 x = 6 , y ° 2 x = ° 4 , y + x = 6 , and y + x = 0 . (Hint: Use a change of variables.)

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4.(6+10 pts.) Consider the iterated integral Z 4 0 Z p 4 ° z ° p 4 ° z Z p 4 ° z ° x 2 ° p 4 ° z ° x 2 1 dy dx dz (a) Change the order of integration to dz dx dy . (b) Change to cylindrical coordinates, and evaluate the triple integral.

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Unformatted text preview: 5.(6+6+6 pts.) For each of the vector &elds below, check whether it is conservative or not. Find a potential function if it is conservative. (a) F ( x; y; z ) = h x 3 ; y 3 ; z 3 + 1 i . (b) F ( x; y; z ) = h e x cos( y ) ; tan ( y ) e x cos( y ) (sec ( y ) & x ) ; xy i . (c) F ( x; y ) = h (1 + xy ) e xy ; e y + x 2 e xy i . 6.(6+12 pts.) (a) Show that I C (2 xy + e x 2 ) dx + (( x + 1) 2 + ln(2 + sin ( y ))) dy = 2 I C ( x & y ) dy where C is the plane curve parametrized as x = cos ( t ) + 1 10 cos 2 ( t ) , x = sin ( t ) + 1 10 cos 2 ( t ) for ± t ± 2 & . (Hint: Use Green&s theorem.) (b) Evaluate the second line integral above directly using the parametrization. (c) What is the area of the region enclosed by C ?...
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