You may have been wondering, why do integrals always have that dx at the end? And what is
that dx, anyway? Well, there are a few answers to that question. Just like with
anything else in math, your understanding of dx will get more and more complete as you
get to higher and higher levels of math. A simplified, incomplete understanding of dx
might serve you best when you're just starting out with integrals, but you'll need a more
accurate understanding as you go along. So let's start with the simplest answer, and
then peel away the first few layers of the onion.
Answer number one: dx is just notation. In other words, the integral symbol, and the
dx, are like bookends that frame the function that's supposed to be integrated. That function
in the middle is called the integrand, by the way. Basically the integral symbol and
the dx just work together to say, integrate whatever's between us.
Answer number two: The dx tells you which variable to integrate. If it's a dx, you're
supposed to integrate with respect to x. If you had a dt instead, then you'd integrate
with respect to t. In other words, the integral symbol tells you to integrate the function,
and the dx tells you to do so with respect to the variable x. That's why the variable
in the integrand usually matches the variable in the dx, or dt, or d anything.
Now you might ask, why do I need the dx to tell me which variable to integrate, when
the function only has an x in it in the first place? Great question! And you're right. In
single variable calculus, the dx, or dt, or d whatever, feels a little irrelevant, because
if there's only one variable in the integrand, then you already know which variable to integrate.
It's pretty obvious that you would integrate f of x with respect to x, f of t with respect
to t, or f of theta with respect to theta. But here's why it's going to matter later.
When you get to multivariable calculus, and you have more than one variable inside the
function, you'll need to know which variable to integrate, and that dx, or dt, or dy, will
be an absolute necessity. As an example, this double integral is telling you to integrate
first with respect to y, and then later with respect to x.
Answer number three, the hardest, but best answer: dx is a differential, which means
that it represents the difference between two things. So more specifically, it represents
the distance between two values of x. But while capital Dx is usually used to represent
any distance large or small, lowercase dx is what we use to specifically represent an
infinitely small distance between two x-values. In other words, the distance between two x-values
that are so close together, you care barely tell them apart.
For some of you this might be yesterday, but think back to whenever you learned different
methods for approximating the area under a curve, like Riemann sums and Trapezoidal rule.
Those rules used a bunch of rectangles, or trapezoids, to approximate the area under
the curve. You just find the area of each rectangle, and then add all those areas together.
Those methods rely on a consistent width for each rectangle. That width is called delta
x, and delta x is just the distance between each x-value.
So what happens to delta x when we want to try to find exact area, instead of just an
estimation? Well, the integral is what's going to give us exact area, but the reason it gives
exact area is because we change delta x to dx when we switch from the sum to the integral!
Where delta x was the width of a rectangle, using dx instead of delta x makes the width
of each rectangle infinitely small. Since the dx is the only thing that makes those
rectangles infinitely thin, it's the only way for us to get to exact area. Which is
why when you take the integral, and you're finding exact area, it always has to be accompanied
by that dx.
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