prac3A

prac3A - a) (10) Compute dxdy in terms of dudV if u = x 2...

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18.02 Practice Exam 3 A 1. Let (¯ x, ¯ y ) be the center of mass of the triangle with vertices at ( 2 , 0), (0 , 1), (2 , 0) and uniform density δ = 1. a) (10) Write an integral formula for ¯ y . Do not evaluate the integral(s), but write explicitly the integrand and limits of integration. b) (5) Find ¯ x . 2. (15) Find the polar moment of inertia of the unit disk with density equal to the distance from the y -axis. 3. Let v F = ( ax 2 y + y 3 + 1)ˆ ı + (2 x 3 + bxy 2 + 2)ˆ be a vector ±eld, where a and b are constants. a) (5) Find the values of a and b for which v F is conservative. b) (5) For these values of a and b , ±nd f ( x, y ) such that v F = f . c) (5) Still using the values of a and b from part (a), compute i C v F · dv r along the curve C such that x = e t cos t , y = e t sin t , 0 t π . 4. (10) For v F = yx 3 ˆ ı + y 2 ˆ , ±nd I C v F · dv r on the portion of the curve y = x 2 from (0 , 0) to (1 , 1). 5. Consider the region R in the ±rst quadrant bounded by the curves y = x 2 , y = x 2 / 5, xy = 2, and xy = 4.
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Unformatted text preview: a) (10) Compute dxdy in terms of dudV if u = x 2 /y and V = xy . b) (10) Find a double integral for the area of R in uV coordinates and evaluate it. 6. a) (5) Let C be a simple closed curve going counterclockwise around a region R . Let M = M ( x, y ). Express c C Mdx as a double integral over R . b) (5) Find M so that c C Mdx is the mass of R with density δ ( x, y ) = ( x + y ) 2 . 7. Consider the region R enclosed by the x-axis, x = 1 and y = x 3 . a) (5) Use the normal form of Green’s theorem to ±nd the ²ux of v F = (1 + y 2 )ˆ out of R . b) (5) Find the ²ux out of R through the two sides C 1 (the horizontal segment) and C 2 (the vertical segment). c) (5) Use parts (a) and (b) to ±nd the ²ux out of the third side C 3 ....
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This note was uploaded on 04/18/2008 for the course 18 18.02 taught by Professor Auroux during the Fall '08 term at MIT.

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