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prac3a

# prac3a - r on the portion of the curve y = x 2 from(0 0...

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MIT OpenCourseWare http://ocw.mit.edu 18.02 Multivariable Calculus Fall 200 7 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .

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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. 2 3 3. Let F vector = ( ax y + y 3 + 1)ˆ ı + (2 x + bxy 2 + 2)ˆ be a vector field, where a and b are constants. a) (5) Find the values of a and b for which F vector is conservative. b) (5) For these values of a and b , find f ( x, y ) such that F vector = f . c) (5) Still using the values of a and b from part (a), compute F vector · dvector r along the curve C such C that x = e t cos t , y = e t sin t , 0 t π . 4. (10) For F vector = yx 3 ˆ ı + y 2 ˆ , find C F vector · dvector r on the portion of the curve y = x 2 from (0 ,
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Unformatted text preview: 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. 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 Mdx as a double integral over R . C b) (5) Find M so that Mdx is the mass of R with density δ ( x, y ) = ( x + y ) 2 . C 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 ﬂux of F v = (1 + y 2 )ˆ out of R . b) (5) Find the ﬂux 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 ﬂux out of the third side C 3 ....
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