By Jane. For example: If y is a function of x, then we sometimes write the derivative of y with respect to x as the following: When we write indefinite integrals, they are written as:. Now, we can see where the notation for the integral comes from.
The bounds of integration from a to b are like the first and last x values for the summation. Health Professions. Law School. Graduate School. Middle School. Related Content November 12, Tips for studying biology.
It even means this in derivatives. A derivative of a function is the slope of the graph at that point. Slope is usually measured as the y difference of two points divided by the x difference of those points:. But the closer these points get the smaller these differences get. Let's start calling them deltas, because the difference between two points is often called the delta of those values.
The deltas get smaller and smaller as these two x,y points get closer and closer. When they are an infinitely small distance apart, then the delta-y and delta-x is shortened to dy and dx:. Having met some poor math teachers myself and some excellent ones, thankfully I can empathize. Can I suggest that you just 'park' your old teacher's suggestion and look afresh - from a new perspective - at calculus?
Before taking calculus, your should have been taught about the limits of functions. Otherwise differentiation and integration would be impossible to understand in the context of algebra and geometry.
I strongly recommend Teach Yourself Calculus by P. Old editions from s are available on Amazon secondhand. After doing a few of these one can set up a set of 'rules' by which different types of functions e.
But we sometimes need to do the opposite, i. To do this we use the previous relation. I was going to suggest a thorough caning for that old teacher but now maybe the old devil wasn't so wrong after all. Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group.
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Viewed 74k times. Sachin Kainth Sachin Kainth 6, 6 6 gold badges 16 16 silver badges 12 12 bronze badges. That's one of the reasons the teaching methods sometimes don't work well. We need to put a "full stop" to this kind of teaching. It's important for really understanding the notation to know that f x dx is the product of f x and dx, and represents an infinitesimally small area.
The dx is not simply a notational delimiter for the end of the integrand i. Show 2 more comments. Active Oldest Votes. Jonathan Jonathan 7, 2 2 gold badges 20 20 silver badges 38 38 bronze badges. And with good reason. 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. Or, when a differential is defined, all of a sudden the dx has a meaning , but then when an integral is being evaluated, the teacher says, "Oh, the dx is just a formality.
Doctor Jeremiah took the question, focusing on the idea of a definite integral: Hi Nosson, Think about it this way: An integral gives you the area between the horizontal axis and the curve.
Most of the time this is the x axis. But say you need a more accurate area. You could break the graph up into smaller sections and make rectangles out of them. Let's use an infinite number of sections. Now our area becomes a summation of an infinite number of sections.
Again, a lot of details are being omitted to keep things intuitive. And we can rename the w variable to anything we want. The width of a section is the difference between the right side and the left side. The difference between two points is often called the delta of those values. So the difference of two x values like a and b would be called delta-x.
But that is too long to use in an equation, so when we have an infinitely small delta, it is shortened to dx. So what the equation says is: Area equals the sum of an infinite number of rectangles that are f x high and dx wide where dx is an infinitely small distance. So you need the dx because otherwise you aren't summing up rectangles and your answer wouldn't be total area.
I'm not sure if f x is to be integrated. I have two theories, but I can't see the point in writing the expression as it is if either of my theories is correct. If 2 is correct, then how does one know when to "stop integrating" i. I have seen this recently in multi-variate calculus , i.
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