ASSIGNMENT 4
1. Compute
ZZ
3x2 y dA,
R
where R is the rectangle [0, 2] × [−1, 0].
2. Let R be the region in the first quadrant bounded by y = 0, y = x and
x2 + y 2 = 1. Compute
ZZ
4
dA.
2
2 2
R (1 + x + y )
Hint. Use polar coordinates.
3. Compute
Z 4 Z
0
√
x
−x
y
e
dy dx
x/2
by reversing the order of integration.
4. Find thepsurface area of the portion of the sphere x2 + y 2 + z 2 = 4 that is inside
the cone z = x2 + y 2 .
5. As the site of the world’s largest single-dish radio telescope, the Arecibo Observatory in Puerto Rico is one of the largest centers for research in radio astronomy.
The huge ”dish” is a paraboloidal reflecting surface 305 meters in diameter and 51
meters deep. What is its surface area?
x2 + y 2
Hint. Find the constant c from the paraboloid’s equation z =
.
c
1