Two mathematics professors, Bob and Steve, are talking in a bar. Bob laments to Steve that the average person just doesn’t understand math anymore. Steve thinks Bob is being too hard on people. When Bob excuses himself to use the restroom, Steve walks over to a pretty blonde sitting by herself and says “When my friend comes back from the bathroom, I want you to make some pretense of walking by us. I’m going to stop you and ask you a question, and I want you to say “x squared”. Got that?”
The blonde looks puzzled: “x…squared?”
“Yes, that’s right.”
The blonde repeats: “x squared…okay, I think I got it.”
A moment later, Bob returns from the bathroom. Steve says “Hey, I bet I can prove that the average person knows math. Take this woman, for example,” he says, motioning to the blonde who, right on cue, is walking past them. “Excuse me miss, so sorry to bother you, but I wonder if you can tell me: what’s the integral of 2x?”
“x squared” says the blonde, and she starts to walk away. Bob is flabbergasted.
Then the blonde turns around and yells “PLUS THE CONSTANT OF INTEGRATION!“
First, I know it’s been a long time since I’ve updated this blog. When I started writing it, I thought I would be able to dig up enough nerdy jokes to fill a football stadium, but I lost momentum as it became more difficult to find jokes that dealt with topics about which I had not already written.
Still, I’m not ready to call “Nerdy Jokes” a done deal just yet. I’d like to continue to update this blog whenever possible, if for no other reason than for my own education and for the entertainment my two or three loyal readers. Thanks: I love you guys!
Now then…integration is a mathematical process from calculus with applications in physics and other sciences. To teach you how to do integration would be far beyond the scope of this blog, so here are some helpful links. Take a few minutes to study these websites, and you should be up to speed on calculus in no time at all!
Integration is the opposite of differentiation, and it’s much easier to understand integration in that context. Plus, understanding what differentiation is will make it easier to comprehend the blonde’s triumphant rejoinder vis-à-vis the constant of integration. So let’s start there.
Consider the function f(x) = x2, whose graph is shown below.
This graph is a parabola; in fact, you might say it’s the archetypal parabola. Every other parabola you can imagine (and I don’t know about you, but I can imagine quite a few parabolas!) is merely a variation on this theme. Some are wider, some are skinnier, some are upside down, but all parabolas have in common a certain set of characteristics; chief among them is that the height of any point on a parabola is directly proportional to the square of the horizontal distance of that point from the parabola’s axis of symmetry.
But pardon me; I digress. Could you find the slope, or slantiness, of this parabola? The answer seems to be no; after all, since the parabola is not a straight line, it does not have a constant slope. But what if I asked you to find the slope of a single point on the parabola? Again, this seems like an impossible task; the slope of any line segment can be expressed as rise over run; a single point, by definition, has neither rise nor run, and so it cannot have a slope.
Fine, fine, but what if I told you to imagine a straight line lying tangent to the parabola at a certain point, say, x = 1. Surely you could find the slope of that line, couldn’t you?
And the answer is: yes, you can find the slope of a straight line lying tangent to a curve at a specific point. That’s where differentiation comes in. When you differentiate a function, or find the derivative, you come up with an expression that gives the slantiness of a line that glances off of a function at a certain point. In the case of f(x) = x2, the derivative is f'(x) = 2x. (You can read the first part as “f-prime of x”. There are many different ways to indicate a derivative.) So if you want to know the slope of a line lying tangent to the graph of f(x) at the point x = 1, you simply substitute 1 for x in the derivative equation, and that gives you the slope of that particular tangent line:
f'(x) = 2x
f'(1) = 2(1) = 2
The slope of line lying tangent to f(x) at x = 1 is 2, which means that this line rises 2 units for every 1 unit it runs. A line tangent to any other point of the parabola would, of course, have a different slope.
If you’re still struggling with the concept of differentiation, here’s a real-world analogy that might help. Say you’re sitting at an intersection when the light turns green. You step on the gas pedal, causing your car to accelerate from rest to, say, 100 km/h (roughly 60 mph, for those not inclined to use metric units). If you were to look at the speedometer at any time while speeding up, the reading of the speedometer would be similar to the derivative of a function. It would tell you how fast you were going right that moment, not ten seconds ago or three minutes from now.
Integration is the opposite of differentiation. Imagine that I gave you the derivative of an equation, and asked you to suss out the original equation. You could use integration for that purpose. If 2x is the derivative of x2, then x2 is the integral of 2x. Right? Riiiiiiight?
Well, almost. See, there are infinitely many functions that have the derivative 2x. Besides f(x) = x2, there’s also:
- f(x) = x2 + 1
- f(x) = x2 – 1
- f(x) = x2 + 8
- f(x) = x2 – 148
- And so on…
Basically, you can add or subtract any number – called a constant – from x2 and you’ll get a new function that has the exact same derivative.
- If f(x) = x2, then f'(x) = 2x
- If f(x) = x2 + 1, then f'(x) = 2x
- If f(x) = x2 – 148, then f'(x) = 2x
- And you get the point…
It’s impossible, given only the derivative of an equation, to integrate and figure out which constant to add or subtract. You need additional information. Neophytes learning about integration for the first time are often told to include a constant of integration – a variable that stands for any conceivable number that might be added to or subtracted from the function. When asked to integrate 2x, the most appropriate response is:
∫ 2x dx = x2 + C
Where C is the constant of integration. It could stand for 4, or -5, or it could even be zero. But we don’t know for sure, so we use C as a stand-in. Unless we’re given further information (say, the coordinate at which the function intersects the y-axis), we cannot determine what the value of C is.
The blonde in our joke proved that she was calculus-savvy enough to run with the mathematics professors – perhaps even more savvy, since Bob failed to mention the constant of integration in his instructions, and she still managed to stick the landing. Good for her.