A calculus problem

[Simon]

I'm filling out my assignment in calculus integration. I have come across I believe a little bit of a trick question. I am being asked to calculate the area enclosed by a sinusoidal curve. Very cleverly they seem to have asked that I calculate the area of the curve Y = sin X between 0 pi and 1.7 pi. Well the integration of the sine function is negative cosine. The 1st trap I could fall into is to forget that the cosine of 0 is one and that I should not forget to carry out that operation even though it involves a 0. So I did my calculation but I cannot understand why the result is 0.412. With my electronics hat on I know that the average area of a full sine wave is 0.707 or one half of the square root of 2. And working it out as though I was trying to calculate an RMS voltage gives me a very different result 0.6 in fact. Then I thought I wonder what happens if I calculate the area of the full curve from 0 to 2 pi. Guess what the result was? 0 so presumably the negative part of the graph is counting against the positive part of the graph. So my 0.412 is actually the top portion of the graph minus the partial bottom portion. So presumably to solve this correctly I need to work out the to portion separately and add them together? So I would need to work out the area enclosed by the curve between 0 Greek pi and one Greek pi and then work out the albeit negative area between one Greek pi and 1.7 Greek pi and add the absolute values together. Is this reason incorrect or should I forget getting a merit in this damn thing and just go for a pass in which case I don't even have to solve this one. Is actually be my electronics knowledge that has saved me from the obvious pitfall.

After all I met a guy the other day who has commissioned me to do paid work for him despite having a HNC in electronics which is what I am doing. He passed his despite blowing his project up so this really is a waste of fucking time isn't it?

[ade]

You do understand it.

0.412 is indeed the "top" part of the curve, minus part of the "bottom" part:



The math itself is easy.  Since you already figured out that the integral is -cos(x),  just calculate the upper minus the lower:

-cos(1.7 pi) - -cos(0)  = -0.588 - -(1) = -0.588 + 1 = 0.412

[Simon]

Yes but they are asking for the area enclosed between the curve and the X axis so I think what they are actually asking is for me to calculate the area of both portions and add them together. The pitfall is if I treat them both together as one curve or I would have to translate the curve upwards to bring it into the positive domain only but I'm not going to mess with that one.

The full question reads: Sketch the curve for y = sin(x) between x = 0 and x = 2 pi, then find the total area enclosed by the curve y = sin x and the X axis between x = 0 and x = 1.7 pi.

[Simon]

actually I give up. I have done enough to get a pass at this stupid thing, at the end of the day it's just about the bit of paper!

[ade]

If that's the case, then the easiest is just think it as the integral from 0 to pi, plus the absolute value of the integral from pi to 1.7pi.



(Since we know that the intercept is at pi, and the area under the curve is negative, we can just subtract the 2nd term):



The left side of the minus sign is just -cos(pi) - -cos(0) = 2.  The right side is then just -cos(1.7pi) - -cos(pi) (which should be a negative number).

[Simon]

yes and if I do it I could get a merit instead of a pass but I'd have to go back and re answer another question that I correctly solved with a quadratic equation instead of calculus, if I use calculus I get a merit. I think I'll leave it on this module. It's taken me fasr too long already.

[ade]

Yeah, in either case, you got the concept exactly right & understand it, which is what's important anyways.

I'm trying to work out some control theory problems and don't remember the math anymore.  So for the past three months I've been going to Khan Academy (https://www.khanacademy.org) every day to brush up my math starting back from Algebra I.  It will take me maybe another year to get back through calculus and differential equations.  Very humbling, esp.  considering I was a math/cs major at one point. 

But I'm going to stick learning day by day until I get my math skills back up, it will be worth it in the long run.  Sal Khan is my hero right now.

[Andy Watson]

I would urge you to practice it - even if there are other methods of getting to the same answer. If you are going to pursue higher levels of qualification in electronics you will find it much easier if you can speak fluent maths. This sort of thing needs to be second nature such that you can concentrate on the electronics bits.

I did type out the answer but I see several other people have beaten me to it

[Simon]

understanding is one thing. passing someone elses concept of understanding is totally another

[Simon]

I would urge you to practice it - even if there are other methods of getting to the same answer. If you are going to pursue higher levels of qualification in electronics you will find it much easier if you can speak fluent maths. This sort of thing needs to be second nature such that you can concentrate on the electronics bits.

I did type out the answer but I see several other people have beaten me to it

try having ADHD! maths requires memory, I can't memorise that sort of thing, I study, forget, get to the assignment question, understand which part I need but can't remember the detail, restudy, do the question, do that for every question, submitt assignment and get a merit. But I still can't do it off the top of my head. I just understand the calculus concepts, don't ask me to work one out unless i can consult a book again.

[ade]

Simon do check out Khan Academy if you haven't.  Khan has helped a ton of people with ADHD learn a lot of different subjects (math, physics, computer programming ,etc.)

And it's all free.  There are no "grades", no deadlines.  Learn at your own pace.  Re-learn any concept anytime.   Jump around to different topics if you'd like.

He's made me a believer out of lifelong learning.   And today with tools like Matlab, Mathematica, or even Wolfram Alpha free on the web, you don't need to do the actual calculations if you don't want to.  Just use your understanding and let the tools do the hard work.

Also I'm still new here but the community at EEVBlog is great.    We can help each other out when get stuck.

[Simon]

I personally don't have a problem with my limited maths skills. The problem I have is passing the goddamn stupid tests. There are lots of people who are good at memorising things and passing tests these people get qualifications and therefore good jobs and often they are shit at those jobs I come across them all the time. There are though those of us that might not be good at maths and at passing exams but are creative in finding solutions and are good at problem solving the sorts of things that are hard to test for. I have in fact said just as you are suggesting to many people as soon as I am past this qualification I will never carry out another detailed mathematical calculation again unless I really need to because I will be using a multitude of tools available. What is the point of knowing how to work out a complex calculus formula when I can just put my formula into a computer and get the result particularly when the objective of getting the result is not to get the result to pass some test but to get the result to use in a higher function of what I am doing. If I spend my life doing maths I will never have time to do electronics so I don't expect to do details maths I've just got to past the stupid tests which thankfully I am doing in the form of assignments at home because if I had to do this in a classroom without any reference to the theory I would file the 1st question. Education system we have come to rely on is a total and utter farce.

[ade]

I agree Simon.  Especially today there's a lot of math that's about mechanically applying some procedure that you have to memorize.  Useless!

The beauty of something like Khan Academy (3rd plug) is that he's trying to help you (and me) get an intuitive understanding of what's going on.  Just like your calculus problem above, you understand it!  Then tools can be used to solve the problems.  You can type "integral of abs(sin(x)) from 0 to 1.7 pi" into wolframalpha.com and it will spit out the correct answer.

The converse to that is... if you don't understand the concepts -- and I mean, really understand the concepts to the point that it's intuitive -- then you can't effectively use the tools.    So Andy Watson is 100% correct too, that understanding of the underlying math concepts needs to be deep that it becomes second nature.

[Simon]

concepts are not a problem, I get those, it's memorizing procedures. Like i did above I did what I thought was the procedure but knew the result was wrong, so worked out why. I won't get a mark for that, I'll get a mark for passing by applying the right procedure. Obviously I was asked to calculate only part of the full sine wave cycle because if I've been asked to calculate the full cycle it would have been immediately obvious what the problem was where as many people will probably calculate the way I initially did and take result as being correct where was I decided because I was not convinced to calculate the area of the full sine curve and then realised what the problem was.

One of the questions was obviously solvable very easily with a quadratic equation, I understood that I can also solve it with minimum and maximums in calculus but solved it before i got to that bit ( I now study a bit then start solving the assignment questions so that I don't keep going round in circles as I described above). Then I got to the bit that explained how to do it,

Basically I can waste a lot of time getting merits and distinctions or I could spend the time earning money running my own business. Like I said some guy with a HNC in electronics is employing me to do what he should be able to do and he admits he understands nothing about electronics. He said he can do logic gates because that's just maths. I rest my case qualifications are a total waste of time.

[pmbrunelle]

I was taught numerical integration before I learned analytical methods.

Everyone knows what addition is from first grade, so integration should be easy...

The introduction into analytical methods right away is backwards, I think.

[Simon]

Well this is supposed to be a HNC course in electronics. Not knowing much about our education system I did not realise this included a heavy dose of higher-level maths. The 1st 3 modules have been cutely named analytical methods for engineers which for many people probably did spell maths wearers as I don't have much of a background in education I thought having signed up and seen the title oh well it must be some light practical use of maths and how to problem solve to get me ready for electronics. The annoying thing is I am already fairly competent in electronics and I don't think I'm going to get much out of these 8 modules. 3 of them are maths one of them is a project you know the thing you can blow up but still pass and by the sounds of it that can be as silly as programming a Rasberry Pi to drive a motor not even been able to select the correct MOSFET or heatsink it properly and still pass. Just 4 of the 8 modules will actually be in electronics therefore I don't actually see what value I HNC poses to any employer yet many ask for it. I have thus far been prevented from getting a proper job in electronics because of my lack of qualifications because I am unable to get through maths although I am slowly staggering through this one and providing nobody wants me to sit in an exam room and recite formulas and procedures I might just manage it. But then meeting people who basically get through the course on their maths knowledge but then have no idea what to do with electronics is disheartening it basically means that once I do have this qualification the skills that I do have will go unrecognised. Between my full-time job and my part-time business I keep coming across people who are partially more qualified than me obviously paid a lot more but who are not exactly very competent at the subject they supposedly studied but they have that all-important piece of paper. To a degree that is why I started my own business so that I wouldn't have people asking all the time what are your qualifications and telling me to get lost. They come they asked for what they want I provided charged them and they go away very happy.

[tggzzz]

With my electronics hat on I know that the average area of a full sine wave is 0.707 or one half of the square root of 2.

No, it is zero.

You are thinking of the power, not the voltage. That involves the "voltage squared", where -V*-V => +V², and it is then interpreted as being equivalent 0.7 etc.

The maths is right, your mistake is in misinterpreting the result in other real-world circumstances.

[Simon]

Yes i did suspect I was not getting that quite right but my practical understanding was enough to forewarn me that the reult may be wrong. In fact yes as I pointed out working it our for 0 - 2pi immediately shows that doing 0-1.7pi would yield a wrong result as 0-2pi gives 0 as it's positive area added to an identical negative area.

[IanB]

Yes i did suspect I was not getting that quite right but my practical understanding was enough to forewarn me that the reult may be wrong. In fact yes as I pointed out working it our for 0 - 2pi immediately shows that doing 0-1.7pi would yield a wrong result as 0-2pi gives 0 as it's positive area added to an identical negative area.

OK, but the question you quoted says this, in your own words:

The full question reads: Sketch the curve for y = sin(x) between x = 0 and x = 2 pi, then find the total area enclosed by the curve y = sin x and the X axis between x = 0 and x = 1.7 pi.

So, did you sketch the curve as the question asks? Because if you did, there would be no need for "practical understanding" or "forewarning". If you sketched the curve it would immediately have shown you that there was a positive part and a negative part. I think you are making too much out of this. There was no trick at all. The question was guiding you to the answer.

[Simon]

I know what a sine curve looks like without sketching it, I just sketched a curve to keep the answer happy, I did not have to do math to know what one looks like I've been seeing sine curves for years before I started this math course. The stuff so far has not really focused on the practical problem solving, it has simply told me how to work out the integrals of various types of function and spent a lot of time teaching me how deal with unsolvable ones by rejigging them and applying various tricks. There has been no real demonstration of practical application of this stuff. It may have been mentioned somewhere way back that if you are working out the area and the curve is both negative and positive but I don't remember reading it. So many people will have fallen for just finding the integral of sine x and applying the standard math shown so far. asking to calculate the area under a full sine cycle would have proved very quickly that you can't integrate 0pi - 2pi and caculate the area all in one go. But if you don't pay attention with 0pi to 1.7pi you can forget how to use your head and just apply all those lovely formula's and methods given in the course. I don't remember an example given of a calculation like this. It has been left for the student to work out.

[Simon]

Sadly there is only a pass, merit and distinction, so unless i do everything required to get a distnction doing this one won't help me, so sod it. I'd much prefer if they wanted a minimum of set questions correctly answered and applied an extra mark for each extra effort, in this way's it's all or nothing, so I do nothing. I don't have time to go back and resolve a question i have already answered because it means restudying thqat section to find the exact method.

[tggzzz]

I know what a sine curve looks like without sketching it, I just sketched a curve to keep the answer happy, I did not have to do math to know what one looks like I've been seeing sine curves for years before I started this math course.

No, but you did need to be brought to the understanding of how the pure maths relates to applications thereof. The question helps that.

Some people intuitively understand the pure maths, others like yourself intuitively understand the applied maths. Both are necessary. The pure maths guides the applied maths. The applied maths gives extra purpose to the pure maths.

That feels like another variant of the irritating "engineers vs technicians" or "doctors vs nurses" threads that are too common on this forum.

The stuff so far has not really focused on the practical problem solving, it has simply told me how to work out the integrals of various types of function and spent a lot of time teaching me how deal with unsolvable ones by rejigging them and applying various tricks. There has been no real demonstration of practical application of this stuff.

Again, that is a pure maths vs applied maths issue. I did both separate subjects for A-level.

[Simon]

Well in this course the material seems to be mostly pure math but then when it comes to the assignment you have to know how to apply it to a practical problem and it's those that get you the higher grades as presumably the end goal is to know how to apply it.