Calculus concept clarification
So in calculus there's integration. There's always that dx at end, which my teacher said is a notation to remind us what is the variable. Then when we learned u substitution, we wrote du = stuff dx. How can a notation be equal to stuff? All this is for indefinite integrals. For definite integrals I completely understand that dx is the limit of width of a rectangle as width goes to 0. So what is dx in the case of indefinite integrals?
November 29, 2014
7 Comments • Newest first
you poor soul. It is ok .. only two more weeks until you reach the calculus-free land. I'll be waiting for you.
[quote=Bob11111]Yea I understand it's sum of areas for definite integral. What does it mean in indefinite integral?[/quote]
indefinite and definite are almost the same thing. The indefinite just gives you the form of the antiderivative and the definite integral is the indefinite integral evaluated at the limits of two points.
[quote=Bob11111]Yea I understand it's sum of areas for definite integral. What does it mean in indefinite integral?[/quote]
Like what other people are saying, dx is short for delta-x (difference in x). dx represents the difference in both definite and indefinite integrals no matter what the limits of the integrals are.
Rephrase: They mean the same thing in indefinite integrals as they do in definite integrals -- which is adding up a lot of small differences in x ( aka dx). It's just that for indefinite integrals, there are no limits to set the ends of the area you're looking for that's why you up with and answer of 'f(x) + C' instead of the value of the area.
Yea I understand it's sum of areas for definite integral. What does it mean in indefinite integral?
the derivative of x is actually 1dx. so you need the dx in the integral otherwise you can't just integrate 1 without the dx. i'm not really sure how to explain. if you take multivariable calculus, you'll realize how important the dx is, since it tells you what you're integrating with respect to, since there can be other variables like y and x.
i think it's representing the change in x when you do sum of an area later
The integral is truly the sum of areas. You are looking for the height (function at x) times the width (dx). What the teacher said is true, the dx tells what independent variable you are using to integrate, but that is not the reason why the dx is there. dx is the width of an infinitesimal length of x+delta x as delta x goes to 0