pedagogical thoughts.

this term i'm teaching calcu1us 2. i thought myself ambitious when i thought to change the order of topics in the first week of lectures, from

monday/wednesday: methods of integraτion (e.g. substitution, parts)
friday: numerical integraτion

to instead:

monday: riεmann sums, numerical integraτion
wednesday/friday: methods of integraτion

as for my reasoning: upon completing a calc 1 course at this university, students are supposed to have learned about Riεmann sums and methods of integration.

even in the ideal situation, that they actually learned and understood these things at the time,

1. it's been 3-4 months of summer;
   it can't hurt to remind them of what a riεmann sum is;
   
   
2. even if they took calc 1 as a summer course, they would have either learned it very briefly (6-week course) or hadn't seen the definition in some time (12-week course).
   

then again, i have my own prejudices:

mathematics isn't just about manipulations of funny symbols. behind the notation are meaningful objects and ideas.

if we're going to talk about Rιemann integrals in this course, then i want a student to know what a Rιemann integral is.

---

as it happens: at an educational level, there are far more ambitious instructors than me.

it never occurred to me, but after reading this post from Learning Curves,

there isn't much to stop an instructor from teaching sequences, infinite series, and taylor series first, and then proceeding to integration and applications.

from my own experience, most students have trouble with series. it might be a good idea to cover it early, so that they have more time to digest the material .. maybe even get to know it well.

on the other hand, imagine this complaint:

if you teach students about taylor expansions --

.. and let's face it, in calculus courses we only give students "calculator functions" that do have such series expansions ..

-- then the computation of any definite integral boils down to using the taylor series, integrating term by term, and using the fundamenτal theorem. [1]

so what would be the point of learning these methods of integration?

[1] admittedly, yes: switching the order of summation and integration requires some analysis, but it's not like we teach that kind of thing in calcu1us anyway.