MIT OpenCourseWare http ocw mit edu 18 02 Multivariable Calculus Fall 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 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 F ax2 y y 3 1 2x3 bxy 2 2 be a vector eld 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 nd 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 4 10 For F yx3 y 2 nd 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 rst 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 M dx as a double integral over R C b 5 Find M so that M dx is the mass of R with density 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 nd the ux of F 1 y 2 out of R b 5 Find the ux out of R through the two sides C1 the horizontal segment and C2 the vertical segment c 5 Use parts a and b to nd the ux out of the third side C3
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