3 ms·

from my perspective, integrals are continuous forms of summations. so Σ(6;n=1) f(n) becomes ∫(6;x=1) f(x) or ∫(6;1) f(x) dx The dx term always confused me, es

by Chris2048 5y ago

from my perspective, integrals are continuous forms of summations. so Σ(6;n=1) f(n) becomes ∫(6;x=1) f(x) or ∫(6;1) f(x) dx The dx term always confused me, especially when it started being used in unfamiliar ways e.g g = (dt+1)/dx - but if you think of it as the step variable/component of a summation term I think it helps.

globular-toast 5y ago
One of the confusing things with integration is the difference between definite and indefinite integrals. What you're describing is a definite integral which is indeed a continuous sum. Indefinite integrals are maybe better called antiderivatives and it's less clear what it's all about. Most people just remember a set of rules for calculating derivatives and antiderivates (like integration by parts, the chain rule etc.). I find this really throws people off as it's like two separate concepts both, confusingly, called integration.
- jamesmaniscalco 5y ago
  They're not totally separate concepts though, no? The definite integral is the difference of two values of the indefinite integral (antiderivative). I suppose if you're talking about solving them computationally, you can calculate/approximate the definite integral by taking the limit of the sum as you reduce the step size, whereas performing the indefinite integral requires a different method (chain rule etc.). But when doing them by hand usually the first step in calculating a definite integral is finding the antiderivative.
  - globular-toast 5y ago
    That's the thing. What tends to stick in people's heads is the bit that hurts. You can tell someone that integration is "just a continuous sum" and you might see a ligthbulb go on. But then the bad news: you still need to find this magical function F(x) and to do that requires intuition and guesswork. Understanding it's "just a sum" doesn't help at all with finding the antiderivative.
    - Chris2048 5y ago
      > Understanding it's "just a sum" doesn't help at all with finding the antiderivative No, but you can still integrate computationally (difference method).
- Chris2048 5y ago
  if definite integrals can be represented as a sum of a finite series, I can't see why an Indefinite integral isn't the sum of an infinite series. You can just take any definite integral and replace it's interval with variables. Now it's just algebraic, much like if you replaced numbers in an equation with variables. > Indefinite integrals are maybe better called antiderivatives but why? I can't see why definite integrals aren't equally better called antiderivatives over an interval. Concepts wrt calculus, and method of solving them are separate things to me - you can understand how gravity works in general, without the specific means of solving orbital equations.

globular-toast 5y ago

One of the confusing things with integration is the difference between definite and indefinite integrals. What you're describing is a definite integral which is indeed a continuous sum. Indefinite integrals are maybe better called antiderivatives and it's less clear what it's all about. Most people just remember a set of rules for calculating derivatives and antiderivates (like integration by parts, the chain rule etc.). I find this really throws people off as it's like two separate concepts both, confusingly, called integration.

jamesmaniscalco 5y ago
They're not totally separate concepts though, no? The definite integral is the difference of two values of the indefinite integral (antiderivative). I suppose if you're talking about solving them computationally, you can calculate/approximate the definite integral by taking the limit of the sum as you reduce the step size, whereas performing the indefinite integral requires a different method (chain rule etc.). But when doing them by hand usually the first step in calculating a definite integral is finding the antiderivative.
- globular-toast 5y ago
  That's the thing. What tends to stick in people's heads is the bit that hurts. You can tell someone that integration is "just a continuous sum" and you might see a ligthbulb go on. But then the bad news: you still need to find this magical function F(x) and to do that requires intuition and guesswork. Understanding it's "just a sum" doesn't help at all with finding the antiderivative.
  - Chris2048 5y ago
    > Understanding it's "just a sum" doesn't help at all with finding the antiderivative No, but you can still integrate computationally (difference method).
Chris2048 5y ago
if definite integrals can be represented as a sum of a finite series, I can't see why an Indefinite integral isn't the sum of an infinite series. You can just take any definite integral and replace it's interval with variables. Now it's just algebraic, much like if you replaced numbers in an equation with variables. > Indefinite integrals are maybe better called antiderivatives but why? I can't see why definite integrals aren't equally better called antiderivatives over an interval. Concepts wrt calculus, and method of solving them are separate things to me - you can understand how gravity works in general, without the specific means of solving orbital equations.