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w 0 f T M55-LE2-001

Mathematics 55 Elementary Analysis III

Second Long Exam Second Semester, AY 2014 -2015

I. Let G be a solid in the first octant bounded below by the cone 2 2

3

x yz

and above by the plane

3z . Suppose that the density at a point ( , , z)x y in G is ( , , ) 2f x y z z .

a. Set up an iterated triple integral that gives the mass of G using rectangular coordinates and

spherical coordinates.

b. Find the mass of G using spherical coordinates.

II. Let 2 2 2

( , , ) , ,x y zF x y z zye xze xye . Find the divergence of F and curl of F .

III. Let 2( , , ) 2 cos , sin ,z zF x y z x y e x y ye .

a. Show that F is conservative by finding a potential function for F .

b. Find the work done by F on a particle that moves on any smooth curve from the point (0,1,0)

to the point (2,0,3) .

IV. Evaluate 2( )C

xy y ds where C is the lower half of the circle 2 2 9x y , described in the

counterclockwise direction.

V. EvaluateC

ydx zdy xdz , where C is the line segment from (0,1,2) to (1,3,6) followed by the line

segment from (1,3,6) to (1,3,2).

VI. Let 2( , ) 2 ,F x y xy xy x . Using Green’s Theorem, evaluate C

F dR , where C is the triangular

path traced in the counterclockwise direction with vertices at the points (0,0), (1,0) and (2,1).

VII. Find the flux of ( , , ) , ,F x y z x y z across the part of the positively-oriented paraboloid

2 21 4 4z x y above the xy-plane.

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