Area, Volume, and Arc Length
Imagine that it's 2:15 in the morning. You are restless and surfing through television channels, and you come across an infomerical about the best knife set you will ever own. With just a phone call, you can own a knife set that will splice an atom for just 5 easy payments of $57.95. But if you call in the next 22 minutes and 37 seconds, you will get the knife set and a dual mechanical backscratcher/hair dryer for only for easy payments of $57.95.
Now that you've bought all of this, you need a good use for the life-changing knife set. Lucky for you, we have a great use for them in calculus. In this chapter, we are 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 can 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 are going to eat him.
We will learn that, by adding up all those bits, we can find of his volume. Although there's nothing for Ornery to be afraid of, he's already warned his friend, Wily the wire. We are 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. With our new-found knife skills, our imaginations are our only limits. We can begin drawing and understanding our own angry blobs, outraged areas and crossed curves.