Name
ID
MATH 253 Honors
EXAM 2
Sections 201-202
Section
1-4
/40
5
/15
6
/15
7
/15
8
/15
Fall 1999
P. Yasskin
Multiple Choice: (10 points each)
2
2
0
y
1. Compute ; ; Ÿx y dx dy.
a.
b.
c.
d.
e.
2
4
6
8
10
2. Compute ;;; 2xy dV over the solid region R given by x 2 t y t x and 0 t z t x.
R
a.
b.
c.
d.
e.
1
70
1
35
2
35
4
35
None of these.
1
2 2
3. Compute ;; e x y dA over the region R in the 1 st quadrant between the circles
R
x 2 y 2 4 and x 2 y 2 9.
a. = e 5
2
b. = Ÿe 3 " e 2 4
=
3
2
c.
Ÿe " e 2
d. = Ÿe 9 " e 4 4
e. = Ÿe 9 " e 4 2
2
2
0
y
4. Compute ; ; e "x dx dy.
2
a. 1 Ÿ1 " e "4 2
b. 1 Ÿ1 " e "4 4
c. 1 Ÿe "4 " 1 4
d. 1 e "4
4
e. " 1 e "4
2
2
5. A cupcake has its base on the xy-plane. Its sides are the cylinder x 2 y 2 4 and its top
is the paraboloid z 6 " x 2 " y 2 . Its density is > 3 gm3 . Find its total mass and the
cm
z-component of its center of mass.
3
6. Find the mass and the z-component of the center of mass of the hemisphere
0tzt
25 " x 2 " y 2 whose density is given by - 1 Ÿx 2 y 2 z 2 .
5
4
7. A cardboard box is constructed with a hinge at the back so that the top, bottom and
back have one sheet of cardboard while the sides and front have two sheets of
cardboard. If the volume is 3 ft 3 , find the dimensions of the box which minimize the
amount of cardboard needed.
5
2
y2
1 between the lines
8. Compute ;; x dx dy over the region inside the ellipse x 16
4
R
y x and y " x in the 1 st and 4 th quadrants.
2
2
HINT: Use the elliptic coordinate system:
2
x 4t cos 2
1
-4
-2
0
2
y 2t sin 2
4
-1
-2
6