# pracfin - Math 32b F05 Practice Final 2005 spherical...

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Math 32b F05 Practice Final 2005 spherical co-ordinates: x = ρ sin φ cos θ , y = ρ sin φ sin θ , z = ρ cos φ , ( x,y,z ) ( ρ,θ,φ ) = ρ 2 sin φ 1. Evaluate 1 0 1 y = x sin( y 2 ) dydx. (Hint: you should first change the order of integration.) 2. A triangular plate has the vertices (0,0), (2,1), and (0,1). Its density at ( x, y ) is y. Find the center of mass of the plate. 3. Let P be the parallelogram spanned by the vertices (0 , 0) , (2 , 0) , (1 , 2) , (3 , 2) . Use the change of variables x = u + v, y = 2 v to evaluate P ( x + y ) dx dy . 4. a)Sketch the level curves of the function f ( x, y ) = arctan ( y/x ) (Hint: this is the polar angle θ of the point ( x, y )), and sketch the vector field f on this same picture. b) Calculate f . 5. Find C F · dr, where F ( x, y ) = ( y 2 + 2 xy ) i + (2 xy + x 2 ) j where C is an arbitrary curve from (1 , 1) to (2 , 3) . 6. a) Give a parametric equation for the plane through the points P = (0 , 1 , 2) , Q = (1 , 2 , 0) , R = (1 , 1 , 1) . Hint the easiest parametrization is r ( u, v
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