Talk:Calculus/Definite integral/Archive 1

Figure 1
Can someone remake figure 1 so that it fits our current definition of the integral? (ie. the partition is into continuous sets of equal measure) I'd do it but I don't know how to make pictures, much less animations, like that. Once this is done, feel free to delete or demote this section, so we don't make the talk more messy.
 * Done --Greenbreen (discuss • contribs) 00:00, 31 May 2011 (UTC)

Reorganization
I went and reorganized the page a bit. I'll go over tonight and update tomorrow morning and hopefully we'll have a nice concise article on integration. --Stranger104 6 July 2005 00:15 (UTC)

Ok, after reading through everything I decided we need to reorganize this articel. I've read through a couple other calc books and I came up with a way of learning Integration that seemed to flow together nicely. heres the organization I was thinking of.


 * Area under curves - define the problem
 * prerequisits
 * Antiderivatives
 * Summation Notation
 * Limits


 * Integrals
 * Origanal Method of solving Integrals
 * properties of Integrals
 * Fundamental Theorem of Calculous
 * Definite vs. Indefinite Integrals

--Stranger104 6 July 2005 02:51 (UTC)

Another suggestion by Juliusross 02:40, 25 October 2005 (UTC)
 * Introduction and notion of area under a curve
 * Definition of Riemann Integrals
 * example (maybe) integration of constant functions from the definition
 * example (maybe) integration of f(x) = x from the definition
 * Basic properties of Integrals (e.g. linearity) (add pictures)
 * Fundamental Theorem of Calculus
 * Antiderivatives in general
 * FTC I
 * FTC II
 * Examples of integration of polynomials
 * Examples of integration of Trigonometric Functions
 * Indefinite Integrals
 * Substitution Rule (seem to be missing from this book)
 * Integration by parts (seem to be missing from this book)

--

This chapter doesn't seem to begin with basic concepts. The two most basic things that I feel the chapter should put first are Riemann Integrals and antiderivatives. I see Riemann Intervals at the beginning, but explained in an extremely wordy, convoluted way. They are explained better later in the article under "Left and Right Handed Riemann Sums." I feel that this section should be moved to the beginning. Antiderivatives are not introduced until ~1/3 through the article, after a formal definition of the integral. Even here, they are only briefly mentioned with no clear explanation.

Additionally, the graphics are not fitting for an introduction. Figure 1 shows intervals of variable size being progressively divided. Equal spacing is the simplest way to start. Figure 2 has no clear point on the curve where the height is determined. Once again, the "Left and Right Handed Riemann Sums" section is more appropriate.

--Rickpock (talk) 06:03, 17 February 2008 (UTC)


 * I realize this is an old topic, but I'm planning on archiving the older discussions on this talk page, and I'd like to address the criticisms raised by Rickpock before doing so. The figure 1 he refers to is the current version's figure 3, which I recently changed to have fixed intervals, so that part has been addressed.  The figure 2 he refers to is the current version's figure 1.  I think the lack of a clear point on the curve where the height is determined is appropriate since the text says that these $$x_i^*$$ points can be any point we want to sample in the given sub-interval.  The current figure 2 (old figure 3) makes this clear, so I think there is no need for change.  --Greenbreen (discuss • contribs) 00:23, 31 May 2011 (UTC)

definition of the integral
What integral do we want to use? The Riemann integral is fine I guess.

For now I have used a variant using partitions of [a,b] into n equal width pieces and let n tend to infinity  This is equivalent for continuous function on [a,b] and a lot simpler. In fact even simpler is to use left and right hand riemann sums and then move onto more general riemann sums (still sticking with equal width partitions). This works fine for continuous functions on [a,b] which is all we are going to need I think.

We can comment/link to other definitions. It is probably a good idea not to bother too much with technicalities.Juliusross 02:41, 25 October 2005 (UTC)

I think we should start with Riemann sums and go from there. We should refer to left, right, upper, lower, midpoint, trapezoidal rule and arbitarty Riemann sums. From uniform and arbitarty partitions. Zginder (talk) 19:40, 29 January 2008 (UTC)

(if you need help)
I'd be glad to help out if you give me a list of stuff todo. I am teaching myself calc as I go but I figured out deferentiation pretty quickly. I also have access to lots of sources as well. --Stranger104 07:55, 16 Jun 2005 (UTC)

Alright heres my ideas for integration so far. first I think we should start out the topic of integration with the discusion of finding area under a function. start off with the more complicated way of finding the area then progress to the fundamental theroem of calculous. what do you guys think?--24.119.127.33 6 July 2005 00:10 (UTC)

Definite integral

 * Mention we are using Riemann sums and link to Riemann on wikipedia
 * explaination of fundamental theorem before giving the proof
 * One example of a positive function and one which takes on both positive and negative values.
 * examples of spotting limits of riemann sums as definite integrals
 * examples using the properties of definite integrals
 * integration of odd and even functions usiing substitution rule

Pictures

 * illustration for constant rule
 * illustration(s) for fundamental theorem of calculus I

Examples and Proofs
What this wikibook could really use is some non-variable examples. I'm teaching myself calculus (we're moving excruciatingly slow in Precalc), and all these new notations have my head spinning, so it would be nice if some examples gave me a clue that I actually understand what all these notations and variables mean.

Proofs are all well and good, but are they really necessary to have here? Perhaps there should be a Integration Theorem Proofs module, because in my Precalc class we learn all the stuff, and then the teacher gives us the proofs to show why. A good analogy would be the quadratic formula. One doesn't need to know how the person got it, and it would just confuse them if they learned the proof before they actually knew the what it meant. Yes, I like to know the proofs, because then I can say "Ah, so that's why this works," but only after I know how to use it.--SimRPGman 05:15, 11 November 2005 (UTC)
 * While "A Geometric Approach" provides an intuitive explanation of the fundamental theorem, it is not a proof at all for at least 2 reasons. It uses the word "infinitesimal" 3 times without explaining what it means. It involves the unjustified (and wrong) manipulations with limits. The other 2 proofs are bad because they use the very subtle intermediate value theorem. All the 3 proofs can be cleaned up and simplified by using the comparison rule. Michaelliv 19:12, 2 June 2006 (UTC)


 * Can you explain what you mean by "non-variable" examples? Do you mean more examples of definite integrals?  If so then I do agree.
 * Also I really think that a book at this level should include the proofs somewhere. Where to put them depends on the context.  For an easy proof which explains what is going on (e.g. some of the proofs of properties of the integral) it is useful to have them in the text.  For more complicated proofs (e.g. Fundamental Theorem, or proof of integration by parts) they should at least be in their own subsection;  and maybe one day it will be useful to move the long proofs to their own pages.  Just my opinion Juliusross 12:14, 11 November 2005 (UTC)


 * By "non-variable" examples I mean examples with definite numbers, similar to the Sales Example on the Differentation page. Concerning proofs: I just feel that the current layout of lead-in explaining the property of integration, showing all the different notations, and then the proof is not ideal.  Why know how to state a property 9 different ways and prove why it works without knowing how it is applied, if even to just random numbers?  All the different "x's," "a's," and "b's" (i.e. variables) can become very confusing.  An example in which I didn't fully understand something until I saw it actually applied with numbers: logarithms.  I was told that if
 * $$\log_ab=x $$ then $$a^x=b$$
 * were the same, but I didn't understand it until I saw that
 * $$\log_28=3$$ because $$2^3=8$$.
 * Basically, it would be nice to relate to something the reader already knows instead of meaningless letters and notations (which would become meaningful after they have been related to).

--SimRPGman 06:50, 19 November 2005 (UTC)


 * Most textbooks have both the general statement and lots of examples. We certainly need more examples (e.g. in the section you mentioned).  Please do add examples to that page if you want to. Whether the example comes before or after the general statement is a matter of opinion, and there is probably no answer that works in all cases.  Juliusross 12:29, 19 November 2005 (UTC)


 * Where would we put them? At the end of each section within this page, or at the bottom of the whole page, or both? Or should there perhaps be another page with example problems, problems, and solutions? In the meantime, can we just put them here on the talk page?--AK7 11:29, 2 January 2006 (UTC)

More numerical examples:they make more sense.
If there are more numerical e.g., al the i's, f's ,x's ,b's ,x's, d's and other variables will make sense to someone who's just learning it. Hard copy mathbooks always have a numerical exapmle under their variable phrase


 * I agree that more numerical examples would be more likely to engage the student learning this material for the first time. I've tried to add some exercises to this page for that reason.  However, it's difficult to give non-trivial examples at this point in the book because the Fundamental Theorem of Calculus has not yet been covered, so very few integrands can be handled easily at this point.  I think the current ordering of topics is logical and any re-ordering of the topics is likely to create new challenges.  --Greenbreen (discuss • contribs) 05:57, 31 May 2011 (UTC)

Add numerical examples here
I'm just adding some basic ones to show what kind of format etc to use.

Numerical Example 1:

Integrate $$\int 2 \cos x $$

Step one: move the 2 to the outside; the dx was implied to start with, now place it in. $$2 \int \cos x dx$$

Step two: take antiderivative of cos x(-sin x), move the - to the outside. $$-2 \sin x + C $$

Argh. I know you're not supposed to edit talk pages, but feel free to have at this. I haven't gotten this TeX down. I actually just started learning it today. I'm not a fan. --AK7 11:43, 2 January 2006 (UTC)

incorrect integral?
small goof, i think. there's a line that claims $\int e^{f(x)} dx = f'(x) \cdot e^{f(x)}$

this would imply that $\int e^{x^2} dx$ is given by an elementary function.

perhaps the integral goes on the other side. thanks!

Please use standard wiki formatting
I've gone through changing all level-one (=) section headers to level-two (==). In some cases this required "demoting" all subsequent headers in a section by one level; in other cases it didn't. In one section I actually changed the structure a bit (see diffs). Please note that the level-one header is always the page title. The section hierarchy added by users should start at level-two and not skip any levels. Similar changes probably need to be made to many other pages in this calculus book. I may work on that later. - dcljr 20:32, 18 May 2006 (UTC)

The Kinetic Energy of molecules at certain temperature
I have a question... How do you integrate this: $$PV=\cfrac{mv^2}{2}\int_{\frac{\pi}{2}}^{\frac{\pi}{2}}\sin^2\theta\cos\theta$$ to get $$PV=\frac{mv^2}{3}\Rightarrow\frac{3}{2}PV=\frac{1}{2}mv^2$$ In other words how do you integrate $$\int_{\frac{\pi}{2}}^{\frac{\pi}{2}}\sin^2\theta\cos\theta$$ to get $$\frac{2}{3}\,$$ Tikai 16:19, 5 May 2007 (UTC)


 * Integration by substitution, take sinx to be u and cosx dx to be du and integrate, remember to change your bounds once you substitute. Alternatively, after you integrate you can unsubstitue, matter of preference. --Cronholm144 10:56, 19 June 2007 (UTC)

Integration needs an Introduction
If you look at the derviative page, there is a clear introduction describing why this tool is useful. There is no such introduction for integration. Why should I care what the area under a curve is? 69.12.151.48 (talk) 06:53, 30 September 2008 (UTC)

Removed caution
I removed the caution box from the "Even and odd functions" section since we are assuming the functions are continuous. I believe that continuity implies integrability, at least for Riemann integrals, but I could be wrong. If I am wrong, feel free to restore the caution. --Greenbreen (discuss • contribs) 12:26, 31 May 2011 (UTC)