Integrals

[AlanEly]

I am new to this forum. I was wondering if someone would be prepared to help me. I managed to just scrape through my Open University Mathematics foundation course quite a few years ago. What let me down, was I was never fully able to understand calculus. Specifically integrals. Does anyone know of an online idiots guide to calculus that might help me. Or I know it is a lot to ask, could someone put an easy to follow idiot-proof guide on here. Thanks in advance.

[Maine farmer]

Sadly, I am unaware of any online "idiot's guide to calculus", but I can recall the days when there were local bookstores that would have had a book on the shelf.  (Oh how I miss the old fashioned bookstores that I loved so much!)  A quick online search yielded some books for sale online.

 

I did well in calculus, but I find myself in need of a refresher course.  I do remember that integration was  differentiation in reverse, but it's been  a good 20 years since I last played around with any equations.

Edited October 23, 2017 by Farming guy

[AlanEly]

Thanks Farming  Guy, I'll check that out. It's about 30 years since I did any real maths. I don't think I had any problem with differentiation. Not sure but I think you  multiplied by the power and deducted one from the power. Was it used to straighten curves on a graph? Though I could be totally wrong. I just could never master integrals.

[Buffy]

So in Googling around, I stumbled upon this thing called Calculus Made Easy by Silvanus P. Thompson F.R.S. which appears to date to 1914, and is in the public domain, so you can just download it.

 

I took only a quick glance at it, and this stuff doesn't change that much (!), and it looks like it's worth reading through if you're on a budget.

 

Now if you're willing to fork over $15 or so, I'd really recommend--don't laugh--The Cartoon Guide to Calculus by Larry Gonick, who's done a whole series of "comic books" on introductory science topics. I got the Chemistry one for my daughter when she was in high school and we both loved it. I've browsed the Calculus one, and it looks good too.

 

There are also the "for Dummies" and "Complete Idiot's Guide to" books, but I can't vouch for them as they vary quite a bit by author.

 

 

I recoil with dismay and horror at this lamentable plague of functions which do not have derivatives, :phones:

Buffy

[exchemist]

Thanks Farming  Guy, I'll check that out. It's about 30 years since I did any real maths. I don't think I had any problem with differentiation. Not sure but I think you  multiplied by the power and deducted one from the power. Was it used to straighten curves on a graph? Though I could be totally wrong. I just could never master integrals.

Your technique is right but not your interpretation. The derivative of a function gives the slope of its curve, its gradient. So with a parabola y = x², the derivative, dy/dx = 2x as you rightly say, but what that means is that at any value of x, the slope of the parabola at that point will be 2x. 

 

Integrals, graphically speaking, give you the area under the curve of the function. If you have the straight line y=2x, the integral of 2x, ⎰2x dx, gives you x² +C where C is a constant whose value you do not know - the so-called "constant of integration".

 

If however you integrate between limits, say between x=0 and x =2, then the area will be the difference between x²+C at those two points, so you can get rid of the unknown C, and you get 2² - 0² = 4.

 

Which checks out: when you draw the line y=2x and look at the area of the right angled triangle you get between the origin and x=2, you have a triangle with a length of 2 and a height of 2x i.e. 4. Area of a triangle is half the base times the height, so yes the area is indeed 4.

 

What you have done in the integration is add up an infinite numbers of thin slices of area, each of height 2x and width dx. When you have more complex curves, calculating the area is less geometrically obvious and the value of integration becomes clear. And then you get into all the other applications of it.........   

Edited October 25, 2017 by exchemist

[AlanEly]

Thanks Buffy. I think that is exactly what I was looking for. It will take awhile to work through but I guess there are no short-cuts. And thank you exchemist, for taking the time and effort to explain it to me. That is much appreciated

[Vmedvil]

They have a error too integrals are not always the solution.

[exchemist]

They have a error too integrals are not always the solution.

Well, you don't mean "error", surely? There are functions that cannot be integrated, but that is rather different, is it not. 

[Vmedvil]

Well, you don't mean "error", surely? There are functions that cannot be integrated, but that is rather different, is it not. 

 

No I mean "Error" which can be calculated via this equation.

 

 

 

What did you think that integrals were perfect there is no way to approximate a summation without error to some degree that is why in calculus 3 you stop using integrals as prone to "Error" which that is what a integral is just a approximation of a summation.

Edited October 28, 2017 by Vmedvil

[exchemist]

No I mean "Error" which can be calculated via this equation.

 

 

Eh? Surely that "error" is the difference between the approximation, done via the trapezoidal rule, and the "correct" value that comes from integration, isn't it? In which case the error is not due to integration, but to alternative approximate methods. 

 

Or have I misunderstood you? 

[Vmedvil]

Eh? Surely that "error" is the difference between the approximation, done via the trapezoidal rule, and the "correct" value that comes from integration, isn't it? In which case the error is not due to integration, but to alternative approximate methods. 

 

Or have I misunderstood you? 

 

No, that error is between the Integral and Summation of the same function. The summation is always correct where as the integral is not. That is what integration is exacting linear approximation of the function which has error.

Edited October 28, 2017 by Vmedvil

[exchemist]

No, that error is between the Integral and Summation of the same function. The summation is always correct where as the integral is not. That is what integration is exacting linear approximation of the function which has error.

I'm a bit baffled. Why then did you copy a formula and explanation relating to the trapezoidal approximation of an exact integral? In the example you gave, it is the "numerical result" that is in error, not the integral. Note that it speaks of the error of the trapezoidal rule. 

Edited October 28, 2017 by exchemist

[Vmedvil]

I'm a bit baffled. Why then did you copy a formula and explanation relating to the trapezoidal approximation of an exact integral? In the example you gave, it is the "numerical result" that is in error, not the integral. Note that it speaks of the error of the trapezoidal rule. 

 

Well, the numerical results of the definite integral will be off by the same amount as predicted by Simpson's rule since by using the function you get a numerical result at a certain point as you use the integral to calculate something. It is the Integrals error at points. The Riemann Integral or definite integral is created by taking the limit of a Riemann summation which has a finite length which still has error in calculating numerical results at a point which still has error in numerical results something that has never been solved completely that is why integrals in all forms are approximations. 

Edited October 29, 2017 by Vmedvil

[exchemist]

Well, the numerical results of the definite integral will be off by the same amount as predicted by Simpson's rule since by using the function you get a numerical result at a certain point as you use the integral to calculate something. It is the Integrals error at points. The Riemann Integral or definite integral is created by taking the limit of a Riemann summation which has a finite length which still has error in calculating numerical results at a point which still has error in numerical results something that has never been solved completely that is why integrals in all forms are approximations. 

I'm sorry but that last statement is one I can't understand, unless you are simply referring to constants of integration. For instance d/dx (x²)=2x. So integral⎰ 2x is x² + C. Surely you are not telling me that that is an approximation, are you?  

[sanctus]

Vmedvil, just post an example with the Riemann integral being wrong/approximate?

[Vmedvil]

Vmedvil, just post an example with the Riemann integral being wrong/approximate?

 

Well, yes the Reimann Summation that the Define Integral is equal to is wrong slightly. http://mathworld.wolfram.com/RiemannIntegral.html

 

Reimann Summation has Error just as the other rules it is just much smaller do you see how the error is .0001 that means at the milisegment of change it will be inaccurate being at the size of a milli the integral distance. so lets say from a to b then that would error out at .0001 of (a + b / 2) size of the object being integrated but taken to an infinity will greatly reduce the error but still it is there. http://www.math.montana.edu/courses/m171/documents/ExampleRiemann.pdf

 

There are several error rules for testing error the most accurate is Simpson's rule if it says there is error then there is error in your integral.

 

http://tutorial.math.lamar.edu/Classes/CalcII/ApproximatingDefIntegrals.aspx

 

Read all the links.

Edited October 30, 2017 by Vmedvil

[Super Polymath]

A key concept (universally) is

because there are no exact wholes in the real world. Edited October 30, 2017 by Super Polymath