This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: ' :7 W155! Gall &m+ Math 241 Summer 2008 Exam 4 Name: Directions: Do not simplify or evaluate unless indicated. Do not evaluate unless indicated! No
calculators are permitted. Show all work as appropriate for the methods taught in this course.
Partial credit will be given for any work, words or ideas which are relevant to the problem. 1. Show that the vector ﬁeld F (117, y) = my 1+ $2312 j is not conservative. [5 pts] 2. A Wire is in the shape of y = x2 for 0 g x S 2. The density at (x, y) is 6(w,y) = 31:. Find the
mass of the Wire. Evaluate. [10 pts] 3. Write down an iterated single integral for the work done by the force ﬁeld F(a:, y) = 1 E — $5
on a particle which travels counterclockwise around the circle x2 + y2 = 1. Do not evaluate. [10 pts] 4. Use Green’s Theorem to ﬁnd the integral f (a: —— 4xy) dx+ 53/2 dy where C is the curve shown
C
in the picture. Evaluate. [10 pts] 5. Find the value of the integral / (y2z3 + 1)da: + 233y23dy + 3my2z2dz
C where C is parametrized by ﬁt) = t3x/t + 1 E— (t2 +t+ 1) j+t 1—9 for 0 S t S 3. Do not simplify. [10 pts]
6. Let E be the cylinder :52 + y2 = 4 between 2 = 0 and z = 3 with the ends sealed with disks
so 2 is closed up. Suppose E is oriented outwards. Draw a picture of Z. Use the Divergence
Theorem to ﬁnd
// (2mi+xzj+z21—€)ﬁ d3
2
Evaluate until you have a triple integral in cylindrical coordinates but do not integrate. [10 pts] 7. Let C be the curve on the sphere $2 + y2 + 22 = 4 directly above the square on the guyplane with corners (0,0,0), (1,0,0), (1,1,0) and (0,1,0). Orient C counterclockwise when viewed
from above. We wish to evaluate / (myi+zyj—z21—c)df
C (a) On the left, draw a picture of the sphere, the square and the curve. Indicate the orientation
of the curve. On the right, draw a surface 2 which has 0 as its boundary. Indicate the orientation of 2 with an arrow. [10 pts]
(b) Use Stokes’ Theorem to convert the integral requested to a surface integral. [3 pts]
(C) Give a parametrization of E. [5 pts] (d) Evaluate this surface integral until you have an iterated double integral. Do not integrate.
[7 ptSl 8. Let 2 be the portion of the parabolic sheet y = :82 between the planes 2 = 0 and z = 3 and
with —2 S :1: S 2. (a) Draw a picture of E. [5 pts] (b) Give a parametrization of )3. [5 pts] (c) Evaluate ff y dS until you have an iterated double integral. Do not integrate. [10 pts]
2 ...
View
Full Document
 Spring '08
 Wolfe
 Calculus

Click to edit the document details