Filling a Container Defined by a Curve
The graphic on the left shows the profile of a circularly symmetric container centered on the vertical axis. Its shape is controlled by moving the locators and selecting either a curved profile or linear segments joining the locators. As you move the slider, the height of the fluid changes. The graph on the right shows either fluid height as a function of volume or fluid volume as a function of height.
Teaching suggestions: As a precalculus exercise, before moving the slider to the right, try to sketch for a particular container shape. First try this with simple containers whose profiles are linear segments and then try curved profiles. For a calculus exercise, indicate on the sketch where the curves are concave up, concave down, or neither. Where are the inflection point(s), if any? What is the relationship between and ? Students might find it easier to sketch if they think of pouring a liquid into the container at a constant rate and then considering .