Why "dx","dy" used in integral?

Question

Why "dx","dy" etc used in integral?Suppose we want to integrate
sin(x),the we write sin(x) dx and then integrate them . but why use"dx" or "dy" etc.

Answer 1

Simple convention..."d" for the Greek letter "Delta" used to denote "change." X and Y originated from the axis of the graph that calculus was originated from.

And to all the blow-hearted braggarts answering this question, can you say, "KIS"?

Answer 2

I think of "dx" as "delta-x", although I'm not certain of the origins in mathematics, or why Newton (or Leibnitz?) decided on this symbol to represent an increment.
The formula is telling you this; you are integrating together all the little (infinitely small) steps of dx to get the total. The "sin(x)" in the formula is any function that depends on the value of x as you are integrating, since it changes (like x=0 to 1) in the formula. So you really have 3 pieces; the dependent bit, the integral telling you how x varies, and the dx that represents all the little pieces you're integrating together.

Hope this helps. The formalism of starting with a dy/dx formula, separating the dy and dx terms, then integrating both sides, is a little more complicated.

Answer 3

Actually, dx and dy are typical infinitesimals. Differentiation or derivative focuses on infinitesimal and integration begins from infinitesimal. For example take infinitesimal, dx, dy, dz, d(theta),...when they are accumulated in an order we get corresponding macro units. Systematic accumulation in calculus is called as `Integration'. Hence when dx is thought off as a typical point of a line and when these points are arranged side by side uni-directionally we get a line. This verbal statement is written symbolically in Mathematics as `Integration of dx = x ' ; in the same way ` int. of d(theta) = theta' and etc. further a line is a typical point of a surface or an area. ` int. of x dx' = (x^2)/2 ; which is actually the area of a right triangle with equal legs and it is also equal to half of the area of the square with side equal to `x'. This process of integration can be continued. To understand integration and defferentation, knowledge and pondering on infinatesimals will be helpful.

Answer 4

None of the above answers are entirely correct. Let me explain:

If we were only integrating with respect to one variable, then you are correct in thinking that it is unnecessary to write in the differential (dx, dy, dz, etc). However, when you have multiple integration such as:

(z+y+2x)dxdzdy

Then in this case you know which integration to perform first. So if you are only integrating with respect to one variable, then it is redundant to write the differential. In my opinion it is unnecessary and stupid. However, as I explained, it's important and required with multiple integrals.

Brenmore,Shasti and Photonic: You ought to read a question carefully before you answer it. The asker is not asking what dx and dy are, he/she is asking why it is used as part of the integral notation. Did someone say KIS? That's right, if you read properly, you may be able to Keep it simple!

Answer 5

Strictly speaking, they are not needed. But they are useful to keep us honest when doing integration by substitution or by parts. Having the dx, du, etc. around makes sure that we use the formulae correctly. Because of this, it is good habit to use them at all times.

The difficulty comes when double or triple integrals are done. For them, the substitution formula is more complicated than just dxdy would suggest. At that point, it is more appropriate to use differential forms, but unfortunately, those are not usually introduced until a differential geometry class.

Answer 6

I believe integral just means the sum of the area from curve to it's axis.
If it is dx then it pretty much means that you're finding the sum of areas from the curve to it's x axis, while dy is the sum of areas from the curve to the y axis.

Integral of sinx dy is (sinx)y (If you are integrating with respect to y, and there are no y in the equation, sinx are treated as constants)
Integral of sinx dx is -cosx

This is the reason why adding dx or dy is necessary.

Answer 7

Suppose if you want to integrate sin(x), we have to specify with which variable we are integrating it. If we are integrating it with x, we use "dx" and the answer is -cosx+constant. If we are integrating it with y, we write "dy". Here we cannot find the answer. That's why we use "dx" or "dy". This is similar to differentiation where we specify with which variable we are differentiating. Bye.

Answer 8

When information is plotted on a graph, the main axis are usually noted as the X and Y axis.

To identify the slope of a straight line, you need two points on that line, and it's calculated as the differences of the y distance (dy) divided by the differences of the x distance (dx).

Putting it simply, with differentiation you can calculate the slope of a line with one point. It's still the difference of the y point as it goes to it's limit divide by the x point as it goes to it's limit.
dy - 'difference of the y distance'.
dx - 'difference of the x distance'.
Integration is basically the reverse of integration.

Answer 9

actually it can be d(anything)...

what dx, dy, d(theta), dz, d(whatever) signifies is a differential increment of the integral... if its a single integral its a one dimensional increment along the axis of differentiation... the double intgral will have differential area and it will be dxdy or dxdz or drd(theta), or whatever, and the tripple integral will have differential volume, dV (which breaks down to dxdydz in cartesian, drd(phi)d(theta) in spherical and so on.)

Answer 10

d(x) is 'with respect to x'
d(y) is 'with respect to y'

in English " integrate the function [sin x] with respect to x"

so when you integrate [sin x] from 0 to pi d(x), you're finding the area under the curve between those two points on the x axis.