Name
ID
MATH 253
Section
EXAM 2
Sections 501-503
Fall 1998
1-8
/40
9
/20
10
/10
11
/20
12
/10
P. Yasskin
Multiple Choice: (5 points each)
2
x
1
1
1. Compute ; ; y dy dx.
a.
b.
c.
d.
e.
"1
3
1
3
2
3
7
6
4
3
2. Find the volume under the surface z
2x 2 y above the triangle with vertices Ÿ0, 0 ,
and Ÿ0, 4 .
"1
3
1
3
2
3
7
6
4
3
Ÿ1, 2 a.
b.
c.
d.
e.
1
1
3. Compute ; ;
0
a.
b.
c.
d.
e.
y
1
y
; x y dz dx dy.
0
1
24
1
12
1
2 2
3
2 2
3 2
4. Find the area enclosed by one loop of the daisy r
=
32
b. =
16
=
c.
8
d. =
4
=
e.
2
sin 42:
a.
1
-1
0
1
-1
2
5. Compute ;
4
=
0
;
=
y2
y sinŸx 2 dx dy.
a. 1
b.
c.
d.
e.
4
1
2
=
4
=
2
3= =
4
6. Compute ;
a. = ln 3
4
b. = ln 3
2
0
;
4"y 2
y
1
dx dy by converting to polar coordinates.
1 x2 y2
8
c. = ln 5
4
d. = ln 5
8
e. = ln 17
4
3
7. Compute ;;;
x2
y 2 dV over the region D bounded by the paraboloid
D
z
a.
b.
c.
d.
e.
9 " x 2 " y 2 and the xy-plane.
4= 3 4
5
= 34
2
= 35
2
2=3 4
2=3 5
8. Compute ;;; z dV over the solid hemisphere H below the sphere x 2
y2
z2
4
H
and above the xy-plane.
a. =
b. 2=
c. 4=
d. 4=
3
e. 8=
3
4
9. (20 points) Find the mass M and center of mass Ÿx , y of the region above the parabola
y x 2 below the line y 4, if the density is > y.
(12 points for setup.)
HINT: By symmetry, x
0. So you only need to compute y .
5
10. (10 points) Plot the polar curve r
1 " cos 2. Show your work.
6
11.
(20 points) Find the volume V and the z-component of the
centroid z of the apple given in spherical coordinates by
>
1 " cos C.
(16 points for setup.)
Hint: In the C-integral, use the substitution u
1 " cos C.
7
12. (10 points) Compute ;; y 2 dx dy over the diamond shaped region R bounded by
R
y
1,
x
y
6,
x
y
x,
y
2x
4
FULL CREDIT for integrating in the curvilinear
coordinates
Ÿu, v where
u2
xy and v 2
y
x.
y2
(Solve for x and y.)
HALF CREDIT for integrating in rectangular coordinates.
0
1
2
x
3
4
8