Name 19 April 2005
Teaching Assistant Hour Test 3
Math 2401E Andrew
Instructions: 1. Closed book.
2. Show your work and explain your answers and reasoning.
3. Calculators may be used, but pay particular attention to instruction 2.
To receive credit, you must show your work. Unexplained answers,
and answers not supported by the work you show, will not receive
credit.
4. Express your answers in simplified form.
1. (25) Find the mass and x coordinate of the center of mass of the rectangle with
vertices (0,0), (3,0), (3,1), and (0,1), given that the density is yxyx
+
=
),(
λ
.
2. (25) Sketch the solid
Ω
described by 1
22 ≤≤+ zyx , and use cylindrical
coordinates to evaluate
(
)
∫∫∫
Ω
+dzdydxyx 22 .
3. (25) a. Evaluate ∫+−
C
dyxdxy where C is the straight line path from (1,2) to (3,0).
b. Find a function f(x,y) for which
(
)
(
)
jiF )sin()cos(1)sin(),( xyxyxyyyxf −++−==∇
and use f to evaluate ∫•
C
drF where C is a curve from
(
)
2
,2
π
to
(
)
π
2,
4
1.
4. (25) A surface S is parametrized by
kjir )sin()cos(2),( 2tstssts ++=
for 10
≤
≤
s and
π
20
≤
≤
t.
a. Convince my that S is a part of a paraboloid about the x-axis by finding a constant
k for which
(
)
22 zykx += .
b. Use the parametrization above to calculate the area of S.
