One should then compute the displacement in terms of the time period. The next step is to explicitly calculate some simple examples to get a feel for what the result is like. Also, it is worth pointing out that using these rectangular approximations is equivalent to finding the area under the graph. Using this intuition, all the symbols make sense, "$\int$" being an elongated "s" for sum, "$dt$" being the size of the thin time intervals, and the bounds of the definite integral being the endpoints of the time period under discussion. Subsequently we can see that if the train's velocity is sufficiently nice we can obtain good upper and lower approximations by choosing sufficiently thin intervals. For each of those intervals we can see that the displacement is clearly the velocity multiplied by the time elapsed, almost by definition of velocity. So we can approximate the real displacement by comparing with trains that have constant velocity over each short time interval. Then we employ the intuition that if a train has higher velocity than another at every point in time, it would travel at least as far in the positive direction. One natural way of expressing that would end up being the derivative of position with respect to time. First we of course have to define velocity, which is the rate of change of position at the point in time, measured by taking smaller and smaller intervals of time around the point. I think that a first introduction to integration should demonstrate what it can be used for in the real world, motivating it by asking how we can find the distance traveled by a train whose velocity changes over time. It's so ancient that it was fixed even before I was born. In such situations, I usually tell the class something like: "I'm not responsible for this notation. As for the use of $x$ in place of $t$, I won't try to defend a convention that notationally identifies a bound variable (in the integral) with a free variable (in $F(x)$). The integrals with various $a$'s as lower limits are some of the antiderivatives of $f$ for some purposes we'll want all the antiderivatives of $f$, and the formulation with $C$ provides these. Why do we omit the limits of integration ($a$ and $t$ above), allowing the role of $a$ to be taken by the $C$ on the right side of the equation, and replacing $t$ as the argument of $F$ with the variable of integration $x$? The use of $C$ can probably be explained thanks to the fundamental theorem of calculus. But then questions arise about the usual notation, which looks like (Different choices for $a$ give different "constants of integration".) So far, the concept and the notation should look pretty reasonable. If I were teaching the indefinite integral to students who already understand the definite integral and are comfortable with the usual notation for it, then I would introduce the indefinite integral as a definite integral with a variable upper limit, i.e., Is there a better way to describe the meaning of the dx (and maybe even the S) here? I tell my students that the S part means "find the anti-derivative", and the dx means "and x is the variable". But, since we're stuck with it, I'd like to know if there is any explanation one can give for the symbols. I think it's a very unfortunate symbol choice. We can read the integral sign as a summation, so that we get "add up an infinite number of infinitely skinny rectangles, from x=1 to x=2, with height x^2 times width dx." Each part of the symbol makes sense.īut with indefinite integrals, the tall skinny S does not mean summation, and the dx has no real meaning. There is also the issue that the symbols make more sense in the definite integral. ("Yep, just find the anti-derivative, that's what we already learned to do when we see this tall skinny S.") If we do indefinite integrals before the Fundamental Theorem of Calculus, the students are likely to think the FTC is not a big deal. I teach the definite integral before the indefinite for a few reasons, one of which is that I want students to recognize that the definite integral means area (not anti-derivative).
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |