Basic Calculus - Help needed

[d&q]

In my Alg 2 class, its the end of the year and we have covered most of Alg 2, so we are just doing trigonometry. We went over graphing the trig functions, inverses, law of sin/cos/tan, etc. Anyway, for extra credit we were supposed to approximate the area of the function y = sin x at the interval [0,π] using Riemann sums. Our teacher said the person who comes up with the closest approximation will get a whopping amount of extra credit. All we had to do was show our steps and etc. Now, me being me, decided to try and find the integral of problem. I already knew the derivative of sin x = cos x, so I could proceed pretty easily. I found the area under the curve to be 2(Which I know is correct). What I'm asking is, could someone please write out the steps for the derivation and the steps for integration? No need to explain it to me, I just need to know the notation.

Thanks in advance.

[Sidoh]

Can you solve it using FTC2 or do you have to do it the long, grueling way?

[Rule]

In my Alg 2 class, its the end of the year and we have covered most of Alg 2, so we are just doing trigonometry. We went over graphing the trig functions, inverses, law of sin/cos/tan, etc. Anyway, for extra credit we were supposed to approximate the area of the function y = sin x at the interval [0,π] using Riemann sums. Our teacher said the person who comes up with the closest approximation will get a whopping amount of extra credit. All we had to do was show our steps and etc. Now, me being me, decided to try and find the integral of problem. I already knew the derivative of sin x = cos x, so I could proceed pretty easily. I found the area under the curve to be 2(Which I know is correct). What I'm asking is, could someone please write out the steps for the derivation and the steps for integration? No need to explain it to me, I just need to know the notation.

Thanks in advance.

I found a way that doesn't involve using a computer, explicitly uses Riemann sums, and doesn't assume the fundamental theorem of calculus (e.g. how to integrate), and hint: is better than if you were to do 100 Riemann sums by hand (which would take hours and is probably a time spending contest the teacher wanted people to enter!).  Since this is for such a whopping amount of credit, I'll let you ask questions instead of giving you the answer, or I'll post the answer here for $50  .

[Sidoh]

You don't have to do 100 reimann sums, you just have to do increasing numbers of reimann sums to see what value they're approaching.  That's pretty easy, but I still think he should let them use FTC2.

[Rule]

You don't have to do 100 reimann sums, you just have to do increasing numbers of reimann sums to see what value they're approaching.  That's pretty easy, but I still think he should let them use FTC2.

No, this is more elegant than just guessing what the sums converge to,  .

[rabbit]

Rule, how about this: you give me $50 and I'll consider letting you post the answer

[d&q]

I know the answer  , I used FTC2(I just didn't know what it stood for), all I want are detailed steps to explain what I did to my teacher.

[Sidoh]

No, this is more elegant than just guessing what the sums converge to,  .

Even though doing so yields the same result?  If you you 3 or 4 instances of the Reimann Sum procedure, it becomes blatant what the answer is.  While I'm sure I'd like your methd more, you need to remember that Deuce is going to have to use methods that an Algebra II student is capable of undrestanding (I'm not saying Deuce isn't capable of undrestanding Calculus, but he hasn't had the class).  The teacher would become highly suspicious.

I know the answer  , I used FTC2(I just didn't know what it stood for), all I want are detailed steps to explain what I did to my teacher.

FTC 2 is really easy to use, but its proof is kind of complicated.  It would be much more legitimate for you to prove your answer by using Reimann Sums, as your teacher has suggested.

[Rule]

No, this is more elegant than just guessing what the sums converge to,  .

If you you 3 or 4 instances of the Reimann Sum procedure, it becomes blatant what the answer is. 

It's good to use intuition, but it can also be very dangerous in mathematics.  Just guessing what something converges to isn't very rigorous, and can get you in a lot of trouble.  For example, how do you know that it converges to 2 and not 1.99999999, or 1.999999999999999999999999999?  It just seems like it would be nice because 2 is a pretty integer.

Another example:  Does the series  1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 .... + 1/1000000 + 1/n  converge as n --> infinity?  Using our intuition we might think it does: the terms get smaller and smaller until they appear negligable.   There are other ways of seeing it though:
The series can be written as (1 + 1/2 + (1/3 + 1/6) + (1/4 + 1/5 + 1/20) + ...  = 1/2 + 1/2 + 1/2 ... + 1/2 = infinity).

Now, does Sum[1/(n²)] converge?

More complicated example: If a function has a directional derivative in every direction at a point a, is it differentiable at point a? 

Even more complicated example: What value does sin(1/z) take in the complex plane as z-->0?

A more advanced example: Do parallel lines ever meet?

Simpler example: What is the integral of sin(x) from -infinity to infinity?

Slightly more complicated example: What is the area under the curve e^(-x^2) from x = 0 to x --> infinity?

You get the idea .

[Sidoh]

It's good to use intuition, but it can also be very dangerous in mathematics.  Just guessing what something converges to isn't very rigorous, and can get you in a lot of trouble.  For example, how do you know that it converges to 2 and not 1.99999999, or 1.999999999999999999999999999?  It just seems like it would be nice because 2 is a pretty integer.

Since sin x is a nice, pretty function, intuition overpowers skeptisism.  Plus, since he knows how to use FTC 2, he knows it's the correct answer.  If we were dealing with more complex functions, I would agree with your reasoning, but since we're not (and he's restricted to using things like the Reimann Sum), ituition is the best tool to use.

Another example:  Does the series  1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 .... + 1/1000000 + 1/n  converge as n --> infinity?  Using our intuition we might think it does: the terms get smaller and smaller until they appear negligable.   There are other ways of seeing it though:
The series can be written as (1 + 1/2 + (1/3 + 1/6) + (1/4 + 1/5 + 1/20) + ...  = 1/2 + 1/2 + 1/2 ... + 1/2 = infinity).

Now, does Sum[1/(n²)] converge?

More complicated example: If a function has a directional derivative in every direction at a point a, is it differentiable at point a? 

Even more complicated example: What value does sin(1/z) take in the complex plane as z-->0?

Simpler example: What is the integral of sin(x) from -infinity to infinity?

You get the idea .

I don't have the time or motivation to work any of those out.  I'm fully aware that using convergance isn't the best solution for every function, but that's not the question (not to mention anyone who thinks that is an malinformed idiot).

The simple one is obvious ... 0.  sin/cos are cyclic functions.  That, of course, implies that you've solved the net change, not the total change, which would be infinity.

[Rule]

The simple one is obvious ... 0.  sin/cos are cyclic functions.  That, of course, implies that you've solved the net change, not the total change, which would be infinity.

Nope, you fell for the intuitive trap!  Infinity is not a concrete number, and "one infinity" may not equal the "other" < for example x - x^2 , lim x --> infinity = -infinity>.  Since sine is cyclic, how do you know where to terminate the point of integration?  Is sin(infinity) =0, 1, -1, 1/sqrt[2], ...?   

To be more explicit,    Integral[sin(x)] (-infinity,infinity) =  -cos(infinity) + cos(infinity) = ? + ?? = undefined, since trig function(infinity) does not converge to any specific point.

This is why intuition must be so carefully applied.  If we establish using the FTC is not acceptable, then we cannot use it to justify a non-rigorous answer.  There is no obvious reason that the integral should converge to 2 rather than 1.999999999999999999999999.  If FTC is the reason, then FTC should be explained, and the Riemann sum calculations would not be valid support.

[Sidoh]

Nope, you fell for the intuitive trap!  Infinity is not a concrete number, and "one infinity" may not equal the "other" < for example x - x^2 , lim x --> infinity = -infinity>.  Since sine is cyclic, how do you know where to terminate the point of integration?  Is sin(infinity) =0, 1, -1, 1/sqrt[2], ...?   

You don't terminate it and you don't evaluate it at infinity.  The total change of the sin x function is infinity.  Good reasoning, bad wording.  That's what I get for trying to do calculus when I'm playing WoW.

To be more explicit,    Integral[sin(x)] (-infinity,infinity) =  -cos(infinity) + cos(infinity) = ? + ? = undefined, since trig function(infinity) does not converge to any specific point.

Learn to use MathType ...

This is why intuition must be so carefully applied.  If we establish using the FTC is not acceptable, then we cannot use it to justify a non-rigorous answer.  There is no obvious reason that the integral should converge to 2 other than 1.999999999999999999999999.  If FTC is the reason, then FTC should be explained, and the Riemann sum calculations would not be valid support.

Are you suggesting that the teacher expects students in Algebra II to use something taught to Calculus students half way through the year?  To me, it seems that the Reimann Sum solution is much more reasonable, especially since using FTC2 can be used to prove it (without actually proving why FTC2 is true).

[Rule]

Nope, you fell for the intuitive trap!  Infinity is not a concrete number, and "one infinity" may not equal the "other" < for example x - x^2 , lim x --> infinity = -infinity>.  Since sine is cyclic, how do you know where to terminate the point of integration?  Is sin(infinity) =0, 1, -1, 1/sqrt[2], ...?   

You don't terminate it and you don't evaluate it at infinity.  The total change of the sin x function is infinity.  Good reasoning, bad wording.  That's what I get for trying to do calculus when I'm playing WoW.

You think the integral of sin(x) from -infinity to infinity is 0, or am I misinterpreting you? By total change you mean absolute value?   

This is exactly the intuitive trap I set.  The curve |sin(x)| is mostly above the x axis, but is always either above or touching the x axis.  Also sin(x) is a periodic function, (e.g. it does not decrease in magnitude at an increasing rate as we approach infinity).  Therefore logically, and (rigorously enough IMO although some pure mathematicians would disagree), the integral of |sin(x)| from -infinity to infinity is infinite.   

Now here is your intuitive error:  by removing the absolute value sign, we have an equal number of identically shaped curve segments above and below the x axis from -infinity to infinity (false premise), therefore
the integral is zero.

Since you're approaching infinity, you are not terminating your evaluation at any specific x value.  Since sin(x) (and cos(x)) do not converge to any point as x --> infinity, you do not know precisely what you are integrating, but you do know that there may not be an equal area to the right of the y axis as there is to the left of it, and that the net area on either side of the y axis may not be zero.

Here is a slightly more rudimentary question that may help you see my point:
What is the integral of sin(x) from x = 0 to infinity?

To be more explicit,    Integral[sin(x)] (-infinity,infinity) =  -cos(infinity) + cos(infinity) = ? + ? = undefined, since trig function(infinity) does not converge to any specific point.

Learn to use MathType ...

LaTeX is so much nicer, and regular text is so much lazier .

This is why intuition must be so carefully applied.  If we establish using the FTC is not acceptable, then we cannot use it to justify a non-rigorous answer.  There is no obvious reason that the integral should converge to 2 other than 1.999999999999999999999999.  If FTC is the reason, then FTC should be explained, and the Riemann sum calculations would not be valid support.

Are you suggesting that the teacher expects students in Algebra II to use something taught to Calculus students half way through the year?  To me, it seems that the Reimann Sum solution is much more reasonable, especially since using FTC2 can be used to prove it (without actually proving why FTC2 is true).

I expect that he should be able to derive the tools necessary to solve the problem exactly, using riemann sums.  It is not a small task for someone who has just taken algebra, although it is very simple and short in my opinion, and does not use any mathematics he is unfamiliar with.  If he doesn't want to start from just algebra though, taking the definition of a derivative (which is really easy to show) as a given, makes the whole problem very easy.

He could simply calculate a few Riemann sums and prove that his estimation is better as the number of Riemann sums used increases, but just saying it obviously converges to 2 isn't good math or good logic.

[Sidoh]

You think the integral of sin(x) from -infinity to infinity is 0, or am I misinterpreting you? By total change you mean absolute value?   

Total change is this:

http://www.sidoh.org/~sidoh/total_sin.gif

Which, in this case, is infinity.

I didn't think enough about the problem when I answered for net change, which I now agree is undefined.

This is exactly the intuitive trap I set.  The curve |sin(x)| is mostly above the x axis, but is always either above or touching the x axis.  Also sin(x) is a periodic function, (e.g. it does not decrease in magnitude at an increasing rate as we approach infinity).  Therefore logically, and (rigorously enough IMO although some pure mathematicians would disagree), the integral of |sin(x)| from -infinity to infinity is infinite.   

I know.

... not the total change, which would be infinity.

Now here is your intuitive error:  by removing the absolute value sign, we have an equal number of identically shaped curve segments above and below the x axis from -infinity to infinity (false premise), therefore the integral is zero.

I know ... I aknowledged the error two posts ago.  Stop getting so involved with your power trip and read my posts.

Since you're approaching infinity, you are not terminating your evaluation at any specific x value.  Since sin(x) (and cos(x)) do not converge to any point as x --> infinity, you do not know precisely what you are integrating, but you do know that there may not be an equal area to the right of the y axis as there is to the left of it, and that the net area on either side of the y axis may not be zero.

Your wording is still quite misleading, even though I understood my error two posts ago.

Here is a slightly more rudimentary question that may help you see my point:
What is the integral of sin(x) from x = 0 to infinity?

I understood the point long ago, for the third time. 

To answer your question: undefined if you're speaking of the net change; infinity if you're speaking of the total change.

LaTeX is so much nicer, and regular text is so much lazier .

Why?

[Rule]

Heh, you said "total change" is infinity, but you didn't comment on the actual integral, and I believe you said
"good reasoning, bad wording," which does not seem to be a concession that you were originally incorrect?

LaTeX is a special typesetting language used especially by scientific and mathematical journals, but is also used to publish books, write reference manuals, document languages, etc.  LaTeX is used as a standard in academic publishing, and the use of anything else is considered unacceptable.  Using LaTeX2HTML, LaTeX2PS, LaTeX2PDF you can also easily generate really nice looking documents where organization of (hyperlinks, bulleting, etc) is automatically done for you.  I bet that any AP exams (in most subjects) you've seen have been written using LaTeX.

Why does it look nicer?  It just does, I guess .  It's more smooth, elegant, spacing is handled better, at the expensive of not having a nice GUI associated with it.  For example, look at the integral you posted, and compare it to ones here: http://arxiv.org/PS_cache/gr-qc/pdf/9309/9309018.pdf.

Edit: No hard feelings .