Why do integrals always have a dx?

There's so much confusion around dx, especially among Calc 1 and Calc 2 students. And with good reason. dx doesn't start to feel relevant until later on, when you start working with multivariable functions, multiple integrals, and differential equations.

But that's no reason not to at least address dx in an introductory way. In this video, we'll talk about a few of the simplest explanations for what dx does, what it tells us to do, and what it represents.

When you're just starting out with integrals, for the most part, it's okay to think about dx as just notation. It just comes with the integral symbol, and you don't need to know exactly why it's there in order to know that you're supposed to integrate.

Once you start changing variables, it's important to know that the dx always tells you which variable to integrate. That's why you'll see dx with a function in terms of x, dt with a function in terms of t, and dy with a function in terms of y. You'll also need to understand that your limits of integration (if you're dealing with a definite integral), will need to match the variable in your function and the variable in your dx/dt/dy/etc.

When you get to multivariable calculus and multiple integrals, you'll need to realize that seeing dx dy at the end of the integral tells you to integrate first with respect to x, and then with respect to y.

But most importantly, you'll want to understand that dx represents the differential, or the difference between two values of x. It's the distance between two values of x. So, when you use a Riemann sum, or trapezoidal rule to approximate the area under a curve, delta x is the width of each rectangle (or trapezoid).

When you switch from these approximation methods, and start using the integral instead to find exact area, that delta x has to change to dx in order to match the integral. Essentially what you're doing is saying that delta x represents a larger width, which is why you can only get an approximation. But dx represents and infinitely small distance, which is why it gets paired with the integral, and allows you to find exact area, instead of just an approximation.

And that's why you always need a dx whenever you're using an integral.