Math 281                                                                      Name: ________________

Test #3: Sections 16.7, 16.8, 17.1, 17.2                          Date  : ________________

 Directions:  Show all work for full credit. Use only Calculus III techniques to complete problems. No TI-Voyage200, 89, or 92 graphing calculators allowed. You may not use notes, textbooks, or any other reference on this test. Be sure to show all of your supporting work carefully, clearly, and in proper mathematical form. Solutions offered without such supporting work will not receive any credit.

 Set up a triple integral representing the volume of each solid given in problems #1 - #3. Use any coordinate system of your choice. Sketch the solid and appropriate projections. Do not evaluate the integral.

 1.  The region in the first octant bounded above by the cylinder z = 1 – y² and lying between the vertical planes x + y = 1 and x + y = 3.

2.   The upper hemisphere given by .

3.    The region bounded by the paraboloid and above by the sphere .

4.        Set up and evaluate the triple integral using cylindrical coordinates that represents the volume of the solid bounded below by the plane z = 0, laterally by the circular cylinder and above by the paraboloid Sketch the solid and appropriate projections.

1. (12 pts) Find the volume under the plane  and over the rectangle given by . Include a sketch of region R.

2. a.   (6 pts) Switch the order of integration and then evaluate. Include a sketch of the region of integration.

b.       (6 pts) Convert to polar coordinates and evaluate. Include a sketch of the region of

integration.

3. (12 pts) Find the volume of the solid bounded by  and the planes z = 0 and

      z = 3 – x. Include a sketch of the base of the solid.

4. (12 pts) Set up but do not evaluate the following double integral for both orders of integration where R is the triangle bounded by y = x, y = 2x, and x = 2. Include a sketch of R.

5. (14 pts) Find the absolute extrema of on the triangle bounded by x = 0, y = 0, and y = 9 – x.

6. (12 pts) Locate all local extrema and saddle points of the graph of the function .

7. Given the function , complete parts ‘a’ – ‘d’:

 a.       (3 pts) Find .

 b.       (3 pts) Find the direction of maximum change in f(x,y) at (-2, 0).

 c.       (3 pts) Find the maximum value of the directional derivative of f(x,y) at (-2, 0).

 d.       (3 pts) Compute the directional derivative of f(x, y) at the point (-2, 0) in the direction of the unit vector that makes an angle of  with the positive x-axis.

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