Math 2210 Section 1 - Spring 2008
Exam III
Answer the questions in the spaces provided on the question sheets. Show all
work to receive credit. Calculators are not permitted. Each question is worth
10 points.
Name:
1
For a cone with base area b and height h, V = bh.
3
1
2
1. Find the value of the following integral by using geometric principles. Do not
integrate directly.
Z 1 Z √1−y2
1 dx dy
√
−1
−
1−y 2
3
2. Graph the region of integration for the following integral. Then, evaluate the
integral using geometric principles. Do not integrate directly.
Z 3Z
0
z/3
0
Z
0
1
1 dy dx dz
4
3. Evaluate the following integral by using geometric principles. Do not integrate
directly.
Z
π/6
−π/6
Z
2/(cos θ)
r dr dθ
0
5
4. Find the Jacobian for the change of variables from (x, y) to (u, v) where u =
3x + 2y and v = x − y.
6
5. Find the mass of the solid S that is bounded above by the sphere ρ = 1 and below
by the cone φ = π/4 with density δ = ρ.
7
6. Find the center of mass of the cube defined by 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 and
0 ≤ z ≤ 1 with density δ(x, y, z) = xyz.
8
7. Find the area of the part of the plane 2x + 3y + 4z = 100 that is above the disc
defined by x2 + y 2 ≤ 1.
9
8. Rewrite the given integral with the order of integration reversed. Do not attempt
to integrate. (Hint: draw a graph of the region of integration.)
Z 1Z xs 3
sin x − y
dy dx
2 ln(xy)
0
x2