The first chapter turned on a light bulb in my head when it said
d which merely means “a little bit of.” ∫ which is merely a long S, and may be called (if you like) “the sum of.”
I am gonna bookmark it and read it to the end so that when my son grows up, I can teach him.
I've often thought that if I ever became wealthy enough to not have to work, I'd like to go back to school and do a better job of it. Looking back, I kick myself for wasting so much of a period of time where I could have been filling my brain with useful knowledge.
Oh well, though - I got enough education to land my first programming job, so it wasn't a total waste. And a lot of fun was had along the way. Hindsight can be a bit illusory.
Me: If x = the beach then dx = a piece of sand on the beach and ∫dx = all the little pieces of sand
4yo: ...and all of the parts of the ocean because that's the whole beach
I'm also excited to do more with him as he gets older :)
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.
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.
No, but you can still integrate computationally (difference method).
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.