Drugstore Cowboy Rug

When I was growing up there was a saying that went like this, "He's a drugstore cowboy". This referred to people who came by life the easy way, rather than paying their dues as a dusty ranch hand.

True confessions. When it comes to math I am a drugstore cowboy. I use computers to figure things out. I rarely write numbers by hand, and then only on the back of a napkin, usually with mistakes of one sort of another. The sheer boredom makes me yearn for a Colorado Rocky Mountain High sung by John Denver at midnight, a fuel-valve too-hard to turn.

I usually have to see a concept illustrated to truly understand it and for me that is the advantage of computer algebra. There is another advantage to computer algebra. I can do as many calculations in an afternoon as a hot-shot mathematician of the past could do in their lifetime. Proceeding this way enables me to cover more ground than the Lone Ranger on a horse named Silver.

Before I get into the central theme, a couple of things. When we find the area under a curve, it is called integration. A lot of time can be wasted discussing this act of cutting rugs to shape and figuring out what we owe the shopkeeper. Integration is idolized more than it should be in science, mostly because it is used as a gate between the math-haves and math-have-knots like me, the drugstore cowboy.

Being a drugstore cowboy I hate this. If you stay with me I promise to rhinestone your math to another level, but if you believe in reinforcing the cultural divide between knowing and knot-knowing, I bid you a fond farewell.

When we find the area under some curve, we can do it from east to west, from sunrise to sunset. Or we can avoid committing ourselves to such limits and remain indefinite. 

- via http://goo.gl/H6mzVM

Finding the area of a little ole bitty piece of ground it is called the definite integral, meaning, I will definitely tell you where we begin and end, start and finish, floor and ceiling.

Before we get all fancy and change the world let's say I give you a line that rises just as much as it runs. Insiders would say this line is just good ole y = x or even f(x) = x. When we find the area between two stations a and b, it is just the area between the horizon line in front of the fireplace where granny has been knitting the throw rug and the outlaw line y = x whose slope is conveniently how the west was one before we ran out of wood when we were building the shed.

In big city speak someone might say, this is the area under the function y = x between hitchin' post a and hitchin' post b.

Pastor Smirkenthorn preaches, "The place where x equals a is a point." and it is, but that point is on a line where x equals a, so y can be anything so I like to think of it as a line. Not to be confused with, "You're a lyin' sack of pooh".

Someone who doesn't get out of the saloon much, might say that x itself isn't even a point since a point requires an x and a y and they would be right. They would say that x is just the lonesome dove we call a scalar.

- van http://goo.gl/bl9PTg

So here we have been promoting x like snake oil, when it is just the lonesome dove. Yes poor x the scalar acting up like a line too big for its britches! Well listen to the neighbors at the saw-mill. We can even think of x equals a as a plane that flies screaming out of the screen cutting our pretty little world into two half-spaces. Some birds will nest in the left half-space, some in the right half-space and some might perch right on the cutting edge of the blade when the creek water is runnin' low on Thursdays.

- via http://goo.gl/t8Zo0O

Now when some stuffy-minded person writes:

They should be writing:

areaUnderCurve(f(x), x, a, b)

The fancy S looking meathook fiddle-hole thing means son, I mean some, I mean sum, we are adding up little dx slices of f(x) dx from x equals a to x equals b. For me this is a little more self-explanatory and easier to understand. It is also easier for computers so they don't run so hot and boil over.

Now let's apply this so I can get on to what I wanted to talk about.

If we have the line where y = x, we can write that as y is a function of x. We make a deal at the table. You give me a card x and I give you back a card y. If for every card with the number x on it, I just hand you back the same card, it is a pretty slow game. When I draw out what happened, well I go over x and up the same amount in y and we have that same line. But we never did say what the blue area was that drove me to drink in the first place. I mean, that gonna turn my brown eyes blue area was between the line and the x axis..

Finishing with the computer's help, because Lord knows I got bigger cattle to rustle we have:

So to find the area of granny's knit throw rug we just multiply a and b by themselves, divide by two and subtract.

We can also do the less definite, I mean indefinite:

and then evaluate that from a to b.

But forget all that, I told you all that to tell you this. What does it mean when I find the area under one function with respect to another function. I mean instead of with respect to just x. I mean instead of chopping up the poor little x-axis into teeny weeny bits? That is what I really wanted to talk about.

Consider. 

The verb, "integrate" got named wrong by some musician who stared at his violin too long.

Integrate means find the

area-under-the-curve-f-of-x-with-respect-to-x-axis

We might call them names to make them feel special, but they really just accept input functions and produce output functions. Now we even write them with fancy notation, but we can also write them like so:

areaUnderCurve(sin(x),x),
areaUnderCurve(sin(x),x,a,b),

Now the reason I wrote this whole silly thing is because I want to understand what it means to write:

areaUnderCurve(x,sin(x)) or more generally

areaUnderCurve(f(x),g(x)) or more generally

areaUnderCurve(f(x),g(x),h(x),k(x)) or analogously

slopeOfCurve(f(x),x) more generally as

slopeOfCurve(f(x),g(x)) The slope of f with respect to g

and also
limitOfCurve(f(x),x, a) being extended to

limitOfCurve(f(x),g(x),a) being extended to

limitOfCurve(f(x),g(x),h(x))

and that is just for single variable calculations. With multiple variables we can find volumes and masses and even find a quiet moment with inertias.