MTH 243/Section2&4

Instuctor: M. Kulenovic

University of Rhode Island

Practice problems for Exam #3
                                                                                                                     Kingston 11/12/2002



1.   Convert the integral

to polar coordinates and hence evaluate it exactly.

2.  Set up but do not evaluatea triple integral that gives the volume of the solid

(a)  bounded above by the sphere  x2  + y2  + z2  = 2  and below by the bowl surface   z =x2  + y2  .
(b)  bounded above by the bowl surface   z = 6 - ( x2  + y2  ) and the cylinder x2  + y2  = 2 and below by the plane  z =0.
 
 

3.  Consider the region W between a cone centered along the positive  z -axis, with vertex at the origin and conaining the point (0,1,1), and a sphere of radius 2 centered at the origin.
(a)  Write a triple integral which represents the volume of the region W. Use spherical coordinates.
(b)  Evaluate the integral.

4.  Evaluate the integral

,
where W  is the region described in Problem 2 (a).

5.  The path of an object moving in the xyz-space is given by  (x(t), y(t), z(t)) = (cos(t) , t2  +2 , 2t + 1).
(a)  At time t = 1, what is the object's velocity ? What is the speed ?
(b)  What is the acceleration at arbitrary moment t ?
 

6.  Find the work done by the force F = (x2  + 2xy, x (y + 1))   acting on a body as it moves along the line segment from (0,1) to (1,2).

7.   Let  F = ( 2xy+x2  , x2  - 4y +ey  )  and let C  be the path from  (0,0) to (0,1) shown in the figure, consisting of a straight line segment and a parabolic arc generated by a parabola  y  =1 - x2 . Find the work

C F dr

(Hint: Show that  F  is a gradient field)

8.   Let  F = (2x y2  - 3y, - y3  + 2 x2 y - 3x).

 (a)   Find a function f (x,y)  such that  grad f = F .

 (b)   Find C F dr  where C  is the oriented curve shown below.

9.  A particle travels along the curve C  given by  r (t) = [cos t, sin t, t]  and is subject to a forceF= (x , z, -2xy) . Find the total work done on the particle by this force for t  between 0 and p .

10.   Let  F = (y - 4 sin( x) +ln(x+ 4), - 2 x + 5 y7  + ey  ). Find C F dr  where C is the oriented curve bounding the region R whose area is 4. R and C are shown in the figure.

(Hint: Use Green's Theorem.)
 
 

11.   Find the flux of the constant vector field F = (2, 1, 4 ). through the triangle with vertices at  (0,0,0),(2,0,0), and (0,4,0) oriented upward.

12.   Let F = (2xy , -  y2  , 2z  ).

(a)   Find a div F.

(b)   Find S F dA  where S  is the surface of the sphere centered at the origin with radius 5, oriented outward.
(Hint: Use the Divergence Theorem.).

13.   Let S be the part of the surface z = x2  + 3y  lying above the rectangle in the xy - plane with vertices at  (0,0), (2,0), (0,2), (2,2)  and oriented upward. Let  F = (y , z  , x ). Find S F dA.

14.   Let  F = (y2  ,4). Let  S  be the portion of the (half) cylinder of radius  4  centered on the  z  axis, whose points (x,y,z)  satisfy  0<=z <= 2 and  y=> 0.  Suppose  S is oriented toward the  z-axis. Find the flux integral S F dA.

15.  Let F = ( xz + 2 z2  , 2y + 1, 2z +   x2  ).

(a)   Find a div F at the point  (2, -1,1).

(b)   Use the Divergence Theorem to calculate the flux of this vector field throgh the unit cube.

(c)   Find the  curl F .

(d)   Compute the work S F dA  where  S  is the upper hemisphere  x2 + y2  + (z-1)2  = 1

(Hint: Use Stokes' Theorem )