why do integrals always have dx next to them?
What does it mean????
- lenpol7Lv 71 month ago
Becaause if there are several 'unknowns' in the integral, the 'dx' indicates which unknown is to be integrated.
e.g. Int(x + a) dx
- MathLv 71 month ago
Not always but mostly. That means with respect to x.
In u substitution method it's in du, which means with respect to u.
- jeffdanielkLv 41 month ago
The dx tells you that you are integrating with respect to x. Sometimes it might be dy or dz or d(theta) or several of them.
For example, when you take the derivative of y = x^3 , you get dy = 3x^2 dx
This is the dx that appears in the integral. It means the differential of x.
- VamanLv 71 month ago
Normally ydx is the area of a rectangle with height y and width. Integral is the total area. Let a and b be limits. This interval is divided in to 10 parts with equal intervals. The total area will be
y1 dx+y2 dx+y3 dx +.... This sum when dx is very small can be written as ydx integrated between the limits a and b. Therefore dx always comes.
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- PuzzlingLv 71 month ago
In Leibniz's notation, an infinitesimal change in x is denoted by dx. The following page does a good job of explaining it rather simply.
An integral can be used to find the area under a function f(x). To do that, you can think of slicing up the area to really narrow slices. The height is nominally f(x) and the width is some small amount Δx. But ultimately you want to make that width infinitesimally small so that the height is exactly the height f(x) at that point. That infinitesimally small width (in the x direction) is what dx represents.
If you remember back to derivatives, you would take the derivative with respect to x.
d(x^3) / dx
So now when you are taking the integral (aka the anti-derivative), that 'dx' is expected to be there. People will sometimes get sloppy with notation and leave it off, but you shouldn't. Similarly, when taking the indefinite integral, people sometimes forget to include the + C. Don't do that either.
I remember a story of a classmate in calculus that was given a 0 on the first test because they had all the right answers, except didn't include + C.