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EXAM
Exam 2
Takehome Exam
Math 2350-02&04, Spring 2008
April 23, 2008
Corrected Version April 28, 2008
- This is a Takehome Exam. You may discuss the problems with others, but write up your own solutions.
- If not otherwise instructed, you can use a calculator to do the integrals, but state exactly what you used the calculator to compute.
- You must show enough work to justify your answers. Unless otherwise instructed, give exact answers, not approximations (e.g.,
2, not 1.414).
- This exam has 8 problems. There are 340 points total.
Good luck!
Problem 1. Let R be the region in the upper half-plane bounded by the circles 40 pts. (^) x (^2) + y (^2) = 1 and x (^2) + y (^2) = 9. Find ¯y, the y-coordinate of the
centeroid of R.
Problem 2. Let R be the region in three dimensional which lies in the first 60 pts. (^) octant and is bounded by xz-coordinate plane, the yz-coordinate
plane, above by the paraboloid z = 2 − x^2 − y^2 and below by the plane z = 1.
A. Set up an iterated integral for finding the volume of R, where the first integration is with respect to y. Evaluate the integral by hand computation.
B. Set up an iterated integral for finding the volume of R, where the first integration is with respect to z. Evaluate the integral by hand computation. Hint: Change the integral with respect to dx dy to polar coordinates.
Problem 3. Let D be the solid bounded below by the cone z = 2
x^2 + y^2 40 pts. (^) and above by the plane z = 4. Find ¯z, the z-coordinate of the
centroid of D.
Problem 4. Let D be the solid region inside the sphere x^2 + y^2 + z^2 = a^2. 40 pts. (^) Calculate the integral
∫ ∫ ∫
D
x^2 z^2 dV.
Problem 5. Let the surface S be the upper hemisphere of the sphere x^2 + 40 pts. (^) y (^2) + z (^2) = a (^2) (i.e., the part of the sphere in the region z ≥ 0).
Find ¯z, the z-coordinate of the centroid of S and the moment of inertia of S for rotation about the z-axis. These are surface integrals, don’t do the triple integral over a solid region.
Problem 6. Let the surface S be the part of the cone z =
x^2 + y^2 that lies 40 pts. (^) in the region 0 ≤ z ≤ 2. Let F(x, y, z) be the vector field
F(x, y, z) = xzi + yzj + zk.
Calculate the flux integral (^) ∫ ∫
S
F · n dS,
where n is the outward pointing unit normal on S.
