Hi good people,
I am learning integration but more specifically, calculating the area under the curve...I understand the formulae and method to do these calculations, except for one thing:
They talk about...I do not know what this is called in English, however, what it is, is that when the curve is plotted, they also draw a line (Interval) somewhere between the lower and upper limits....I presume it's referred to as "integral with zero length"..I do not know. could someone please explain to me what this "line" is and also how is it calculated?
I have searched the web trying to find something but no luck. Thank you very much!
I'm not sure, but I think this might be something known as the "average value"
For instance......let us find the area of the curve between the function 1 - x^2 and the x axis
The curve will intercept the x axis at x = - 1 and x = 1
So.....using symmetry, the area is
2* ∫ 1 - x^2 dx = 2 [ x - (1/3)x^3 ] = 2 [ 1 - 1/3] = 2 [ 2/3] = 4/3 units^2
Note that we can construct a rectangle with the same area if we let the width range from x = - 1 to x = 1 = 2 units = the interval width
So....the height of this rectangle must be (4/3) / 2 = 4/6 = 2/3
See this image :
Note that the rectangle ABCD has the same area as the definite integral
To always find this average value we can calculate this :
[ area of integration ]
_________________ = height of rectangle = average value
thank you for replying, but that is not it....the line they draw is actually a very narrow rectangular, and it stretches from the x axis upwards to where it meets the graph. I know it is not anything to do with symmetry since it is plotted anywhere (of course calculated).
the book explains how in the past before Integration was developed, small rectangles were used to get an approximate area size. So the area would be filled with rectangles as narrow as possible (x values), and then were the areas of each rectangle calculated and then added together. The area of each rectangle is lb, so since the rectangles are positioned vertically, the length would be y and the breadth would be \(\Delta x\). The greater this value, the less accurate the final answer. (Total Area)
Then they explain that integration makes the \(\Delta x\) value to be zero, thus we have millions of very thin rectangles (with lengths (y) and zero breadths) of which the areas need to be calculated and added together, hence the introduction to integration. Now, one of these "strips" are called "a representative strip"....please bear in mind I translated directly from Afrikaans...so this may not even be the true name for it.
The question is: Show the "representative strip" you are going to use to calculate the area. Then the second question is to use integration to actually calculate the area size.
If nothing makes sense, I will understand....Thanx for trying though...
Ah....you a referring to the concept of the "Riemann Sum"
Here is a very good tutorial on this....it explains it much more clearly than I can :
Thanks Chris, that should help Juriemagic to understand the rectangle idea.
The rectangles can be represented in a number of different ways.
They are estimates of the area under the curve and in certain cases they can give exact areas.
I'd just like to talk a little more about this.
You know how the integral sign looks like a stylized S ?
Well that is not a coincidence.
That S stand for sum.
Consider definite integrals where there is a number at the bottom B and a number at the top T and it is dx
The integral will will give the sum of the areas of the rectangles that are under she curve between B and T.
The rectangles will be evenly spaced along the x axis between the lowest value B and the highest value T
The width of the rectangles is the difference between 2 adjacent x values. ie dx d stands for difference.
For example if two adjacent x values are 3 and 4 then the difference between them is 4-3=1 This is the width of the rectangle.
ONLY with integration the difference between the x values approaches 0
So a definite integral give the sum of an infinite number of infinitely thin rectangles and hence gives the accurate area under the curve.
I am not sure if you can understand what I am saying when I have not any diagram to show you.
Try drawing what i have described.
If you want me to try and explain better ask and I will try. :)
thank you so so much for this. To be honest, yes I do know exactly what you are talking about. The book I have explains all of that in quite detail, and I remember seeing the term "Riemann Sum" somewhere. They are just not very clear on how to determine the position of the "rectangle" in the graph. I have also looked at the site CPhill has provided, but again, I read, but don't understand.
Anyways I have had my lesson with the students and we finished a question paper on this topic, and I saw that nowhere were there questions regarding the Riemann Sum...so I guess it's not all that important to know then.
Thank you all for the help you have given....always appreciated!
I understood what Chris was presenting but i do not know any of it as the Reiman's sum, is a term used more in the US. idk.
I am not surprised that you did not understand what I was saying. It needs diagrams and proper explaination and then it needs a lot of time to assimilate the ideas.
No one will understand it all on first pass. It kind of gets absorbed with use, at least it did for me. (I am still absorbing some of it )
Maybe next time you can upload a pic and we can explain what it is trying to demonstrate.
I should have shown you this site in the first place. Maybe you will understand it better than what I was trying to say.
Actually it is almost identical to what i was trying to say only it is said better. :)