Download 1. Find the volume of the region between a sphere... 2, in the region of R

1. Find the volume of the region between a sphere with radius 1 and a sphere with radius
2, in the region of R3 bounded by x ≥ 0, y ≥ 0, z ≥ 0
2. Evaluate the integral
√
√
2
2
2
Z1 Z1−y Zx +y
xyzdzdxdy
0
0
x2 +y 2
3. Compute the integral
Z Z Z p
x2 + y 2 dV
W
Where W is the region inside the cylinder x2 + y 2 = 1 with x ≥ 0, y ≥ 0, z ≥ 0 and
below the surface z = 2xy
4. Evaluate the integral over the region R which is bounded by the lines y = x, y = x−2, y =
−2x, y = 3 − 2x
ZZ
(3x + 4y)dA
R
Hint: Use the linear transformation x = 1/3(u + v) and y = 1/3(v − 2u) to evaluate the
integral.
5. Compute the integral
Z Z p
3
y 2 /x dA
D
2
over the region Dpdefined by 0 ≤ xp/y ≤ 1 and 0 ≤ y 2 /x ≤ 8. (Hint: use the change
of variables u = 3 y 2 /x and v = 3 x2 /y and its inverse transformation x = uv 2 and
y = u2 v)
6. The implicit equation for a cone is x2 + y 2 − z 2 = 0
(a) Find a parametric equation for the portion of the cone that lies within the region
−1 ≤ z ≤ 1.
(b) Find a downward facing normal vector.
(c) Find the tangent plane at the point (−1, 0, 1).
7. Compute the area of the surface x + z = 1 above the region −1 ≤ x ≤ 1 and −1 ≤ y ≤ 1.
8. A spider has built a circular web with radius 1 located on the yz plane and centered at
the point (0, 0, 1). Near the spider web, the number of insects passing through the point
(x, y, z) each hour is given by δ(x, y, z) = 4z/π. If half the bugs which pass through her
web are caught, how many bugs does the spider catch per hour?