What does dx actually mean? Because calculus teachers really put two letters at the end of an integral and act like the rest of us received a briefing. 😭 Are we multiplying d by x? Is it a number? Can I leave it off? Why does everyone else seem so relaxed about this? The annoying part is that it makes MUCH more sense when somebody shows you what the whole expression is doing. Start with a graph. You want the area between the curve and the horizontal axis. For the graph in this video, everything is above that axis. So you split the region into thin rectangles. Each rectangle needs TWO measurements: HEIGHT × WIDTH. The function gives you the height. But a height on its own is not an area. A three-metre-tall wall could be one metre wide or a hundred metres wide. You still need the other measurement. For a finite collection of rectangles, that little width is written Δx. Read it as “a change in x.” Now multiply each height by its width and add the areas together. Make the rectangles narrower. Use more of them. The staircase follows the curve more closely. The definite integral gives the limit of those sums as the widths approach zero. That is where the picture of all those little widths connects to the dx at the end. Important: this does NOT mean “make every rectangle zero wide and somehow add the zeros.” You take the limit of the TOTALS. More rectangles, smaller widths, a total approaching a definite value. And the x in dx is doing a job too. It tells you which variable you’re integrating with respect to. Which is a very academic way of saying: WHICH THING ARE WE LETTING CHANGE? Look at this: ∫ 2x dx = x² + C Here, x is the integration variable. We’re looking for a function whose derivative with respect to x is 2x. x² does that. Add any constant and its derivative is still 2x. That’s why the + C is there. Now keep the 2x in the middle but change the ending: ∫ 2x dt = 2xt + C That answer assumes x is independent of t. In other words, x stays fixed while t changes. So 2x behaves like a fixed number in this calculation. It’s the same reason integrating 6 with respect to t gives 6t + C. If x happened to be 3, then 2x would literally be 6. Same expression in the middle. Different variable doing the changing. Different calculation. Those two letters were apparently not just there for decoration. Fantastic. Would have loved to know that earlier. One more distinction that saves a LOT of confusion: Changing ONLY dx to dt is different from changing the name of the variable EVERYWHERE. ∫₀³ 2x dx = ∫₀³ 2t dt = 9 Those are the same calculation with different letters. The limits, the function and the integration variable still match. But ∫ 2x dt keeps x in the expression while asking us to integrate with respect to t. That’s why the assumption about x staying fixed matters. And the numbers above and below the integral sign? They give the starting and ending values of the integration variable. For our example, from x = 0 to x = 3: ∫₀³ 2x dx = [x²]₀³ = 9 − 0 = 9 That is 9 square units of area. You can also check it as a triangle: half × base 3 × height 6. So the notation is telling you what you’re adding, which variable you’re working with, and where you start and stop. An integral without bounds asks for antiderivatives. An integral with bounds gives a definite total. Related ideas, different outputs. You don’t have to become fluent in every symbol before you’re allowed to ask what any of them mean. “Why is that there?” is a completely reasonable maths question. Save this for the next time dx appears at the end of a problem like you two have already met. Which maths symbol did your teacher introduce without explaining? 👇 #calculus #math #LearnOnTikTok #integrals #GravityAssist - @gravityassistus