1. Find the volume of the region between the plane x + y + z = 24 and the paraboloid z = x² + y².
 
 
 
2. Find the coordinates of the center of mass of a triangular lamina with uniform density and vertices at (0,0), (a,0), and (b,c).
 
 
 
3. Suppose that when the Portuguese sailors let their dogs and pigs loose on the island of Mauritius the dodo population there immediately headed to the south end of the island, and thus was distributed such that at the north end of the island there were 0 dodos per square mile, and at the south end 120 dodos per square mile. The island is shaped roughly like the lower leaf of the three-leaved rose r = 50sin(3).

a) Find the total dodo population of Mauritius (to the nearest dodo, of course).
b) Find the average dodo population of Mauritius.
 
 
 
4. Suppose the temperature at a point (x,y,z) in a room which measures five meters (from 0 to 5) along the x axis by six meters (from -3 to 3) along the y axis by 3 meters (from -3 to 0) along the z axis is given by the function .
 
a) Produce a graphic representation of the temperature in the room
b) What sort of physical situation might produce such a temperature distribution?
c) What is the average temperature in the room, to the nearest tenth of a degree?
d) If we adjust the function to, what is the average temperature in the room, to the nearest tenth of a degree?
 
 
 
5. [From Howard Anton's Calculus, 3^rd edition, p. 1079] The portion of the surface

between the xy-plane and the plane z = h is a right-circular cone of height h and radius a. Use a double integral to show that the lateral surface area of this cone is .
 
 
 
Extra Credit:  Find the surface area of the region x^2/3 + y^2/3 + z^2/3 = 1, or at least a good approximation or good bounds for it.