Download Test Four Review Sheet

MATH 238 CALCULUS 3
FALL 2009
Exam Four Review
Be able to do these things and you're golden.
The numbers following each concept are good exercises to look at from Schaum’s Chapter 10
Explain the general form of the FTC and how the two new versions we've seen meet this form
Identify (based on a picture or on a function) whether or a vector field is conservative
Basic (old-fashioned) line integrals [Work word problem] 35, 36/37, 38, 40
Evaluate a vector line integral using the FTC for line integrals 48, 49/50, 72
Explain the repercussions of the FTC for line integrals (path independence, etc.)
Evaluate a line integral using Green's Theorem 42, 43, 47
Standard method of evaluating surface integrals
Some extra practice integrals follow.
EX1



for F ( x, y )  xy 2 i  yj and



for F ( x, y, z )  2 xyi  x 2 j  k
 F  dr
C is the line from (0,0) to (1,1)
C
EX2
 F  dr
and
3
 
4

C is r (t )  t 2 i  t 3 j  tk for t  [0,1]
2
C
EX3 Find the work done in moving an object from (2, 0) to (1, 0) along x 2  4 y 4  4 through the force

field F  y 2 i  2( x  1) yj . (0)
EX4
 (e
2y

i  (1  2 xe2 y ) j )  dr

for C : r (t )  tet i  (1  t ) j t  [0,1] (e 5  1)
C
EX5
 (2 xy z
3
4

i  3x 2 y 2 z 4 j  4 x 2 y 3 z 3 k )  dr

for C : r (t )  ti  t 2 j  t 3 k t  [0,2] (2 20 )
C
EX6
 ( xy i  x
2
2

yj )  dr C is the triangle with vert ices (1, 0), (0, 1), (0, 0) oriented positively (1/6)
C
 (2 xy
2
 arctan( x 3 )) dx  (2 x 2 y  ln( y ) )dy
C
EX7 C is the arc of the parabola y  x 2 (1,1) to (2,4), followed by the line segment from (2,4) to (0,4),
from (0,4) to (0,1), and from (0,1) to (1,1)
EX8
 
F
  dr

for F ( x, y, z )  2 xyi  x 2 j  k
and

C is r (t )  t 2 i  t 3 j  tk for t  [0,1]
C

EX9

  F  dS for

F ( x, y, z )  xze y i  xze y j  zk
S
which is in the first octant
EX10
 
F
   dS for
and
S is the part of the plane x  y  z  4
 323 

F ( x, y, z )  xi  yj  z 4 k
and
S is the part of the cone z  x 2  y 2
S
beneath th e plane z  1 with downward orientatio n
3 
2