Suppose you have a choice of two coffee cups of the type shown, one that bends outward and one inward, and you notice that they have the same height and their shapes fit together snugly. You wonder which cup holds more coffee. Of course you could fill one cup with water and pour it into the other one but, being a calculus student, you decide on a more mathematical approach. Ignoring the handles, you observe that both cups are surfaces of revolution, so you can think of the coffee as a volume of revolution.

1. Suppose the cups have height , Cup A is formed by rotating the curve about the y - axis, and Cup B is formed by rotating the same curve about the line x = k. Find the value of k such that the two cups hold the same amount of coffee. 

2. What does your result from Problem 1 say about the areas A1 and A2 shown in the figure?  

3. Based on your own measurements and observations, suggest a value for h and an equation for x = f (y) and calculate the amount of coffee that each cup holds.

OP y x = k h A1 A2 x = f(y) k