Review sheet for test 4 (v1).pdf - Math 241 Review sheet...

Math 241 Review sheet for Test 4 November 12, 2022
1. Let C be the curve in three-dimensional space parametrized by r ( t ) = ⟨ 2 cos t, 2 sin t, t ⟩ for − π ≤ t ≤ π .
(a) Find the mass of a thin wire in the shape of C , if the density function is ρ ( x, y, z ) = x + z + 10.
(b) Suppose that a particle moves along C starting at r ( − π ) = ( − 2 , 0 , − π ) and ending at r ( π ) =
( − 2 , 0 , π ). Find the work done on the particle by the force F ( x, y, z ) = y i − x j .
2. Let F ( x, y ) = 3 y
2
− 2 x, 6 xy , and let C be the linear path from ( − 1 , 0) to (0 , 1).
(a) Evaluate
Z
C
F · d r directly. (Do not use potential functions.)
(b) Show that F is conservative by finding a potential function for it. Use a systematic method to find
your potential function.
(c) Evaluate
Z
C
F · d r using your potential function from the previous part.
3. Let C be the triangle in the xy -plane with vertices at (0 , 0), (1 , 0), and (0 , 1), oriented counter-clockwise.
(a) Evaluate
I
C
2 y
2
dx + 2 x dy directly as a line integral.
(b) Evaluate the line integral in the previous part by using Green’s Theorem and then evaluating a
double integral.
4. Let S be the surface parameterized by r ( u, v ) = ⟨ v cos u, v, v sin u ⟩ for 0 ≤ u ≤ 2 π and 0 ≤ v ≤ 1.
(a) Mark the picture of S below.
(b) Evaluate the surface integral
ZZ
S
y dS .
5. Let Z be the part of the graph of z = g ( x, y ) = 2 −
p
x 2 + y 2 which lies above the xy -plane in R
3
. Find
the surface area of Z .
6. The region D defined by 0 . 03 < x
2
+ y
2
< 1 . 3 is
shown at right. Within this region are three curves
A , B , and C . Each curve starts at (0 , − 1) and ends at
(0 , 1). Suppose that F ( x, y ) = P ( x, y ) i + Q ( x, y ) j is a
differentiable vector field defined on D with the prop-
erties
∂P
∂y
=
∂Q
∂x
,
Z
A
F · d r = − 1, and
Z
C
F · d r = 2.
(a) True or false: The region D is simply connected.
(b) True, false, or cannot determine: F is conservative.
(c) Find
Z
B
F · d r . Choices: − 3, − 2 . 5, − 2, − 1 . 5, − 1,
− 0 . 5, 0, 0 . 5, 1, 1 . 5, 2, 2 . 5, 3
Instructor: Matthew C. Russell

7. Show that the line integral 2 x sin y dx +
(
x
2
cos y − 3 y
2
)
dy is independent of the path C , and evaluate
it if C is any path from (1 , 0) to (0 , 2).
Z
C
8. A vector field G is plotted at right.
(a) Choose the formula for G .