MIT OpenCourseWare http://ocw.mit.edu 18.02 Multivariable CalculusFall 2007 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.��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 33. Let F~= (ax y + y3 + 1)ˆı + (2x + bxy2 + 2)ˆ be a vector field, where a and b are constants. a) (5) Find the values of a and b for which F~is conservative. b) (5) For these values of a and b, find f(x, y) such that F~= ∇f. c) (5) Still using the values of a and b from part (a), compute �F~· d~r along the curve C such C that x = et cos t, y = et sin t, 0 ≤ t ≤ π. 4. (10) For F~= yx3ˆı + y2ˆ, find �C F~· d~r on the portion of the curve y = x2 from (0, 0) to (1, 1). 5. Consider the region R in the first quadrant bounded by the curves y = x2 , y = x2/5, xy = 2, and xy = 4. a) (10) Compute dxdy in terms of dudv if u = x2/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 = x3 . a) (5) Use the normal form of Green’s theorem to find the flux of F~= (1 + y2)ˆ out of R. b) (5) Find the flux out of R through the two sides C1 (the horizontal segment) and C2 (the vertical segment). c) (5) Use parts (a) and (b) to find the flux out of the third side
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