Area, Volume, and Arc Length
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How much space is left under the bed? How much area is under the trajectory of the red angry bird? You can calculate these areas using integrals.
We don't mean integrating the polar bear population into the Antarctic Circle. Learn how to do some simple integration using polar coordinates.
Here's a more link with a few more complicated examples of integration with polar coordinates. It turns out that the penguins wanted friends, after all.
You know how to calculate the area under the angry bird's trajectory in space. Now you can figure out just how far he had to go to take out that greedy pig.
How far do you throw that marshmallow to get it to your friend's mouth? This video will teach you. With some practice, you'll have a neat parlor trick.
Do you get the waffle cone or the regular cone? The answer is simple: the one that holds more ice cream. Make out at the soda shop by learning how to find these volumes.
Games and Tools
Make a bundt cake with a parabola. Make an ice cream cone with a triangle. See how to do it here.
We can also easily make Frisbees with integration. You can visualize how to revolve simple geometries around the origin to make simple geometries with this tool.
The Egyptians and the Mayans made the pyramids. It'd probably have been much easier if they had this tool to see what they were doing.
If you can dream up a shape and write an equation for it, you can revolve it around an axis using integration to make an even neater geometry. The next time an art major says you can't do what they do, show off with this link.