010.104-2006-2-mid

# 010.104-2006-2-mid - Honor Calculus2 Midterm Exam 2006 10...

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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 ...
View Full Document

## This note was uploaded on 09/09/2009 for the course MATHMATICS 010-102 taught by Professor Cho during the Spring '09 term at Seoul National.

Ask a homework question - tutors are online