Exam 1 Math 205 Fall 1998 - University of Michigan



Final Exam Calculus 3 Winter 2013

Name: ___________________ This is a closed book exam. You may use a calculator and the formulas handed out with the exam. Show all work and explain any reasoning which is not clear from the computations. (This is particularly important if I am to be able to give part credit.) Turn in this exam along with your answers. However, don't write your answers on the exam itself; leave them on the pages with your work. Also turn in the formulas; put them on the formula pile.

1. Consider the integral (R( ( x2 dxdydz where R is the region that is inside the sphere x2 + y2 + z2 = 4 and above the cone z2 = x2 + y2.

a. (3 points) Draw a picture of R. Also draw a picture of the cross section S of R that one obtains by intersecting it with a half plane of constant (.

b. (6 points) Convert the integral from Cartesian coordinates to spherical coordinates. You should write the integral in spherical coordinates as an iterated integral including the limits of integration on each integral.

c. (6 points) Evaluate the iterated integral in part b.

2. (14 points) Find where C is the quarter circle in the xy plane centered at the origin going from (1, 0) to (0, -1).

3. (14 points) Find where F(x,y,z) = -yi + xj + k and C is the helix r(t) = (cos t) i + (sin t) j + 2tk from (1, 0, 0) to (1, 0, 4().

4. a (11 points) Let F(x, y, z) = (3x2 + sinz + yz) i + (xz + y) j + (x cosz + xy + ez) k. Find a function f(x,y,z) such that the gradient of f(x,y,z) is F(x,y,z).

b. (4 points) Suppose the force F(x, y, z) in part a acts on an object while it moves along the straight line from (2, 0, 1) to (1, 2, 8). Find the work done by the force on the object as it moves along this path.

5. (14 points) A particle starts at the point (-2, 0) and moves along the x axis to (2,0), and then along the semicircle y = to the starting point. As it moves along this path it is acted upon by the force F(x,y) = x i + (x3 + 3xy2) j . Use Green's theorem to find the work done by the force on the particle as it moves along this path.

6. (14 points) Find the divergence and curl of the vector field F(x, y, z) = ). Simplify your answers as much as possible.

7. (14 points) Find (S( y dS where S is the surface z = (x3/2 + y3/2) as x varies over the interval 0 ( x ( 1 and y varies over the interval 0 ( y ( 1.

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