Name
ID
MATH 311
Exam 3
Section 502
1.
1
/15 3
/15
2
/30 4
/45
Fall 2000
P. Yasskin
(15 points) Compute ;; x dx dy over the region
2
R
2
x 2 y 1 between the lines
16
4
x
x
y
and y "
in the 1 st and 4 th quadrants.
2
2
HINT: Use the elliptic coordinate system
1
inside the ellipse
x 4t cos 2,
y 2t sin 2.
-4
-2
0
2
4
-1
-2
1
1
2.
(30 points) The diamond shaped region at the
right is called an astroid. It is the graph of
x 2/3 y 2/3 1
and it may be parametrized by
x cos 3 t
-1
0
1
y sin 3 t.
-1
a. Find its arc length.
b. Find its area.
HINT: First find one quarter of the length.
HINT: What does Green’s Theorem say about the integral >Ÿ"y dx x dy ?
2
ds for F
Ÿz 2 " Ÿsin x e cos x , Ÿcos y e sin y , 2xz along the helix
3. (15 points) Compute ; F
rŸ2 Ÿcos 2, sin 2, 2 for 0 t 2 t 2=.
is a conservative vector field. Then find a potential OR change the path.
HINT: Check F
3
4.
(45 points) The spiral ramp at the right
may be parametrized as
Ÿr, 2 Ÿr cos 2, r sin 2, 2 R
for
0trt2
and
0 t 2 t 3=.
a. Find the tangent plane at the point Ÿx, y, z Ÿ"1, 0, = . Give its equation in both parametric form
and non-parametric form.
4
b. Compute
;; x 2 y 2 dS
over this spiral ramp.
dS over this spiral ramp if F
Ÿxz, yz, z 2 .
c. Compute ;; F
5