MAT 203 Final, Fall 2005

1. Suppose that the temperature at each point of a thin metal plate is given by the function: The temperature function has four stationary points. a. Find them, and fill out the table below related to the second-partials test. Show your work. Point Type b. What are the coldest and warmest positions along the unit circle (Lagrange method) 2. Let T be the solid bounded above by the hemisphere and below by the paraboloid a. Sketch the two surfaces and by considering the traces of the surfaces in the xz − plane show that they intersect at the level z = 1. Also, sketch the projection of the solid T onto the xy-plane. b. The flux out of the paraboloid is given by where R is the projection of the solid T onto the xy- plane. Compute the integral. 3. Let T be the "ice cream cone" bounded below by the half-cone , and above by the unit sphere x2 + y2 + z2 = 1. a. Set up triple integrals for the volume of T in (i) rectangular, (ii) cylindrical, and (iii) spherical coordinates. Justify your limits of integration. b. Compute the volume of T. 4. Consider the double integral below: a. Sketch the region that gives rise to the double integral above. b. Change the order of integration, and evaluate. Explain your reasoning 5. For the integral Solve the following: a. Sketch the solid T that gives rise to the triple integral below b. Sketch the projection R of T onto the xy-plane c. Evaluate using cylindrical coordinates. 6. Using spherical coordinates, find the volume of the solid on the first octant bounded above by the cone z2 = x2 +y2, below by the xy-plane, and on the sides by the sphere x2 +y2 +z2 = 4 and by the other two coordinate planes. Express each of the boundaries of the solid in spherical coordinates.