Boiling Cauldron: Introduction Now let’s ﬁll the cauldron from our example with water and light a ﬁre under it to get the water to boil (at 100◦ C). Let’s say it’s a cold day: the temperature of the air outside the cauldron is 0o C. How much energy does it take to boil this water, i.e. to raise the water’s temperature from 0◦ C to 100◦ C? y 70oC 100oC x Figure 1: The boiling cauldron. The temperature of the water is not the same at each level in the kettle. At the bottom of the kettle, where you’re heating it up, it’s at its highest tempera ture — 100 degrees Celsius. At the top, it’s going to be, say, 70 degrees Celsius. The temperature is varying in height; if it varies linearly the temperature at height y will be 100 − 30 a y degrees Celsius. The total amount of heat you need to add is going to be temperature times volume, and some places will get more heat than others. The base of the caul dron will be hottest, but it also has the least volume. The cauldron is widest at the top, so we have more water to heat at that level. At each horizontal level, the temperature is constant, so we’ll use horizontal rectangles in this calculation. We’ll revolve these short horizontal rectangles about the y-axis to get disks, calculate the amount of heat needed for each disk, then integrate that value with respect to dy.