> ... I know that he's confusing indefinite and definite integration.
Yes he is. The real solution is to banish the term "indefinite integral" from the language, and substitute "antiderivative". We should also not use the integral sign for antiderivatives, but unfortunately there is no well known alternative. :-(
Using the same terminology and symbols for antiderivatives as for (definite) integrals is confusing, as here. It also makes the Fundamental Theorem of Calculus -- a very profound fact if there ever was one -- appear to be something trivial about getting rid of limits on an integral sign. But consider: one can compute a definite integral using an antiderivative ... whodathunkit?
Isn't it weird how awful math terminology is? Even the word "derivative" is an awful choice. The special and wonderful thing about a derivative is that it's the rate at which another function changes, not that it's derived somehow.
And why is the verb for "take the derivative" "differentiate"? Both words make sense individually to describe things -- df/dx is derived from f by taking differences -- but it doesn't make sense that we use both words.
I had an abstract algebra professor, in my first year of grad school in math, who referred to operators D that satisfy the product rule D(fg) = D(f) g + f D(g) , which are not necessarily defined in terms of a limit, as "derivations". This is a natural generalization of the derivative, and apparently "derivation" is the correct technical term, but for a week or so I thought he was just not speaking English properly.
Yes he is. The real solution is to banish the term "indefinite integral" from the language, and substitute "antiderivative". We should also not use the integral sign for antiderivatives, but unfortunately there is no well known alternative. :-(
Using the same terminology and symbols for antiderivatives as for (definite) integrals is confusing, as here. It also makes the Fundamental Theorem of Calculus -- a very profound fact if there ever was one -- appear to be something trivial about getting rid of limits on an integral sign. But consider: one can compute a definite integral using an antiderivative ... whodathunkit?