010.104-2006-2-mid - Honor Calculus2 Midterm Exam 13r 15r 4...

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Unformatted text preview: Honor Calculus2 Midterm Exam 2006 10 21{ 13r 15r 4 Z 9 < V: s2: M K+ c a Z @\ %V IU lǣ k k( 100). \ 1. (10 pts) Let F(x, y, z) = (sin x cos y, sin x sin y, cos x) be a vector field defined on R3 and c(t) be a flow line for F. Find the length of the curve c(t) for 0 t . 2. (10 pts) Let f (x, y, z) = exy + cos z be a function and F = xyi + yzj + zyk be a vector field defined on R3 . Then compute div( f F). 3. Prove if correct or disprove by showing a counterexample if otherwise. (a) (7 pts) The vector field F(x, y, z) = - r (r2 + 1)3/2 is not the curl of a C 2 -vector field, where r = xi + yj + zk and r = r . (b) (8 pts) Let f be a function defined on a rectangle [a, b] [c, d]. Suppose that the following iterated integral exists b d f (x, y) dy dx. a c Then the function f is integrable on the rectangle [a, b] [c, d]. 4. Evaluate the following integrals. 1 cos-1 y (a) (7 pts) 0 1 0 1-x2 sin(sin x) dx dy. (x2 2 -1 - 1-x + y 2 ) 3 dydx. 2 (b) (8 pts) 5. (15 pts) Find the volume of the region bounded by the surface (2x + y + z)2 + (x + 3y + z)2 + (x + y + 5z)2 = 1. 6. (15 pts) Find the center of mass of the solid bounded below by the sphere = 2 cos and above by the cone z = x2 + y 2 . (Take the density function (x, y, z) = 1). 1 1 7. (a) (10 pts) Determine whether the iterated improper integrals 1 1 -1 -1 y dydx and x+1 -1 -1 y dxdy converge or not. x+1 D (b) (10 pts) Let D = [-1, 1] [-1, 1]. Show that through the definition. y dA diverges by following x+1 ...
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