1. [Stewart, Problems Plus, p.830, #8] The plane x + y + z = 24 intersects the paraboloid z = x² + y² in an ellipse. Find the highest and lowest points on the ellipse.
 

2. [Inspired by Judith V. Grabiner's "'Some Disputes of Consequence': Maclaurin among the Molasses Barrels," from Social Studies of Science 28/1 (February 1998) 139-68] In 1735 the great British mathematician Colin Maclaurin "wrote a 94-page memoir to the Scottish Excise Commission, explaining how to gauge, with a single dip of a dipstick, the amount of molasses in the barrels in the Port of Glasgow." [p. 139] In this treatise he proved several surprising theorems to the general effect that the difference between the frustum of the solid produced by revolving a conic section around one of its axes and an approximating cylinder matching the radius at the midpoint of the frustum depends only on the height of the frustum. In particular, he proved that for a paraboloid of revolution the volume of the frustum is the same as the volume of the cylinder. Use a double integral to express the volume of a frustum of a paraboloid of revolution and show why this is true.
 

3. Maclaurin also showed that the difference between a frustum of a right circular cone and the corresponding cylinder is one-fourth the volume of a similar cone, with the same height as the frustum and with diameter one-half the difference between the upper and lower diameters of the frustum. Use a double integral to express the volume of a frustum of a right circular cone and show why this is true.
 

4. Suppose the temperature in degrees Fahrenheit at a point (x,y) 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 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?
 

5. a) Produce a graph of the surface x^2/3 + y^2/3 + z^2/3 = 1.
    b) Write an integral which would give the volume of the region inside the surface from part a.
    c) Compute the volume of the region from part b.