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Unformatted text preview: Math 3C — Sample Exam #3 Laney College, Spring 2011 Fred Bourgoin 1. Find the volume of the solid above the paraboloid z = x 2 + y 2 and below the halfcone z = radicalbig x 2 + y 2 . 2. Find a formula for a vector field in which all vectors are of unit length and per pendicular to the position vector at that point. 3. Compute each integral. (a) W is the region above z = 0, below z = y , and inside x 2 + y 2 = 4. integraldisplayintegraldisplayintegraldisplay W yz dV (b) D is the triangle with vertices (0 , 0), (1 , 1), and (0 , 1). integraldisplayintegraldisplay D 1 1 + x 2 dA (c) R is the region in the first quadrant bounded by y = 0, y = √ 3 x , x 2 + y 2 = 9. Use polar coordinates. integraldisplayintegraldisplay R ( x 2 + y 2 ) 3 / 2 dA 4. The positions of two particles are given by vector r 1 ( t ) = ( 1 + t ) vector i + (4 t ) vector j + ( 1 + 2 t ) vector k and vector r 2 ( t ) = ( 7 + 2 t ) vector i + ( 6 + 2 t ) vector j + ( 1 + t ) vector k. (a) Do the paths intersect? (b) Do the particles collide? 5. Describe the surface parameterized by x = s + t, y = s t, z = s 2 + t 2 , where 0 ≤ s ≤ 1 and 0 ≤ t ≤ 1. 6. The velocity of a flow at the point ( x, y ) is vector F ( x, y ) = vector i + x vector j . Parameterize the path of motion of an object in the flow that is at the point ( 2 , 2) at time t = 0. 7. Evaluate integraldisplay 3 integraldisplay √ 9 − z 2 − √ 9 − z 2 integraldisplay √ 9 − y 2 − z 2 − √ 9 − y 2 − z 2 x 2 dx dy dz . 1 8. Give an iterated integral you would use to evaluate integraldisplay R f dA , where R is the region pictured below. You do not need to evaluate any integrals for this problem. EC. Let P be the parallelogram with vertices (0 , 0), (0 , 1), (2 , 4), and (2 , 5). Compute the integral using the change of variables u = x / 2 , v = y 2 x ....
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 Spring '08
 HELENIUS
 Cone, Vectors, dr dθ, dθ, Laney College

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