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calc quiz - slices until we reach the height of the...

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An integral part of calculus is slicing things up and adding them together in order to get an exact value for things we don’t have equations for. This idea is applicable to a variety of things, including volume, mass, and other properties. However, it is perhaps volume that is the most straightforward. For example, a cylinder is a solid that is made up of lots of slices. There are two types of slices: horizontal and vertical. Assuming that the cylinder is upright, it makes much more sense to slice it horizontally so that we have many circles, as opposed to vertically, in which we obtain a shape that we have no equation for. When slicing, it is best to find a shape that is easy to work with, such as the basic shapes. To find the volume of this cylinder, we will add up an infinite amount of circular
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Unformatted text preview: slices until we reach the height of the cylinder. Any case will involve adding up the slices to find a quantity; in this case it is volume, but it could easily also be mass. So to add up all these slices, we find the volume of each individual slice and add them together. But because we want these slices to be infinitely thin, we will take the limit of these slices as the thickness goes to zero. This limit-sum directly translates into an integral. The limit as the thickness goes to zero of the sum of all of our slices means exactly the same thing as the integral of the volume of each slice from 0 to the height. By definition, the integral is the sum of a bunch of infinitely small pieces. Then of course, solving the integral, gives the total value-in this case, the volume of the cylinder....
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