Imagine that it's 2:15 in the morning. We're restless and surfing through television channels, when we come across an infomerical about the best knife set we'll ever have the opportunity to own. With just a phone call, we can own a knife set that will splice an atom for just 5 easy payments of $57.95. But if we call in the next 22 minutes and 37 seconds, we'll get the knife set and a dual mechanical backscratcher/hair dryer for only four easy payments of $57.95.
Now that we've bought all of this, we need a good use for the life-changing knife set. Lucky for us, we have a great use for them in calculus. In this chapter, we're going to learn to use knives to slice things up into little pieces. It's up to you to find something to do with that hairscratcher thing-a-ma-jig.
We already know how to cut the area under a function into slices that look more-or-less like rectangles:
Just like we would slice a zucchini, we can chop any weird 3-D blob into tiny bits that look more-or-less like washers. He's not too happy about that:
If we make the slices small enough, we can break a curve into tiny bits that look more-or-less like straight lines:
After chopping our ambiguously shaped, angry blob–his name is Ornery–into tiny bits, we can find how much blob we get in each slice. No wonder he's not happy. He thinks we're going to eat him.
We'll learn that, by adding up all those bits, we can find his volume. Although there's nothing for Ornery to be afraid of, he's already warned his friend, Wily the wire. We're coming after him next.
While this chapter will help us hone our knife skills, this is not a chapter that will teach us to assemble a masterpiece of a dinner. We won't evaluate integrals ourselves. We will be set up to discuss bigger and better things than the area under a curve, though. With our new-found splicing skills, our imaginations are our only limits. We can begin drawing and understanding our own angry blobs, outraged areas, and crossed curves.
Wyzant: Area Under a Curve
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.
Simple Integration with Polar Coordinates
We don't mean integrating the polar bear population into the Antarctic Circle. Learn how to do some simple integration using polar coordinates.
Integrating with 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.
Finding Arc Length
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.
Volume of Solids
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.
Visualization Tools for Integration of Solids
Make a bundt cake with a parabola. Make an ice cream cone with a triangle. See how to do it here.
Visualizing Disk and Washer Integration
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.
Visualizing More Solids
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.