Anyone care to explain to me what dx (or any other variable besides x) is in an integral? I do well in calculus (so its not like im asking for homework help) but when I think about it I don't know if i understand what dx really is
I know integration can be used to find area under the curves (which can be used with real applications) by summing up rectangles. The way I saw it was that since you're summing up an infinite number of rectangles with height f(x), each rectangle would have a width of deltaX. But I still don't think I see why it's dx.
What also confuses me is that when doing something like u-substituion with a function in terms of x, when you differentiate the function you get du/dx = whatever, and since you want to substitute dx in the original integral with du you can use dx as an operator to isolate it
Nathan Roberts
dx is an infinitesimal unit of x. it comes from the definition of a derivative where we increase a function by the smallest possible amount to see how it changes
Thomas Gray
So would that explain why if you have, say, a function for velocity v(t) in m/sec, you would have integral from a to b of v(t)dt, dt would be in seconds and the seconds would cancel out, leaving you with displacement?
Levi Rivera
Look up integration by first principle. When you use integration to find the volume of a sphere you are summing the volume of an infinte number of cillinders from 0 to h. You find the area of a cross sectional circle and then times it by dx to find the volume. Do this an infinite number of times and this sum can be rewritten as an integral.
In short, the dx is the 'height' of the cross sections.
Jayden Robinson
When you think back to summing up those rectangles, you used a change in x along the x axis for the base of the triangle when calculating. That distance is called delta x. "dx" is its equivalent. It really isn't much more than a symbol showing you that you're making steps of infinitely small size along the x direction instead of large "delta x" steps. When you do other integrals and see things with dy and dx in your integration, you'll see that it is very important to keep which axis you are integrating along straight, so dx is a big deal. But in early integration, it kind of falls away on its own and everything is convenient so nobody really talks about it.
Juan Garcia
>times it by lad...
Caleb Allen
It's literally just notation.
Grayson Ortiz
yeah my bad, old habits die hard
Jordan Reyes
for example, surface area of a cone
Aiden Cooper
as user already said, it's just notation. I may be wrong on this one, but when this was invented (Leibniz?), they actually treated the dx's as objects on their own, but it was non-rigorous, it just "somehow" worked. the "somehow" is now well understood - manipulation with differentials is actually just chain rule and linearity of the derivative operator (which are basic theorems in calculus) in a fancy disguise. this notation is extremely convenient so it is still used but really there is no such thing as "multiplying by the dx".
