Why does integration give the area under a curve?

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I can understand the process of differentiation and how that's used to find a tangent by using limits, but have absolutely no idea how integration gives the area under a curve Is there a similar, more intuitive explanation for integration?

I know that an integral can be defined as the limit of a Riemann sum, but how do we even know that the limit of the Riemann sum converges to the integral in the first place? I think that I saw an example using the fundamental theorem of calculus, but it was pretty confusing and I don't think I understand it that well

Thanks for helping!

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Hope that helps, apologises for the handwriting


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(Original post by majmuh24)
I can understand the process of differentiation and how that's used to find a tangent by using limits, but have absolutely no idea how integration gives the area under a curve Is there a similar, more intuitive explanation for integration?

I know that an integral can be defined as the limit of a Riemann sum, but how do we even know that the limit of the Riemann sum converges to the integral in the first place? I think that I saw an example using the fundamental theorem of calculus, but it was pretty confusing and I don't think I understand it that well

Thanks for helping!

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(Original post by majmuh24)
I can understand the process of differentiation and how that's used to find a tangent by using limits, but have absolutely no idea how integration gives the area under a curve Is there a similar, more intuitive explanation for integration?

I know that an integral can be defined as the limit of a Riemann sum, but how do we even know that the limit of the Riemann sum converges to the integral in the first place? I think that I saw an example using the fundamental theorem of calculus, but it was pretty confusing and I don't think I understand it that well

Thanks for helping!

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Out of interest do you know how the trapezium rule works?
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(Original post by m4ths/maths247)
Out of interest do you know how the trapezium rule works?
Yep, I understand all the numerical integration methods (including Simpson's rule), but just can't understand why an integral is exact.

I know it's something to do with the limit of a sum, but I don't know why this limit converges to the integral of a function.

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JAIYEKO
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(Original post by majmuh24)
I can understand the process of differentiation and how that's used to find a tangent by using limits, but have absolutely no idea how integration gives the area under a curve Is there a similar, more intuitive explanation for integration?

I know that an integral can be defined as the limit of a Riemann sum, but how do we even know that the limit of the Riemann sum converges to the integral in the first place? I think that I saw an example using the fundamental theorem of calculus, but it was pretty confusing and I don't think I understand it that well

Thanks for helping!

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It's relatively easy

Solve this

If two planes have the equation

2x - 2y + z = 24
x + 3y + 4y = 8

And they meet at line L, find the Cartesian equations for line L

Once you worked that out, you'd find your original question easy

And I know the answer because I've worked it out right now?!?
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(Original post by JAIYEKO)
It's relatively easy

Solve this

If two planes have the equation

2x - 2y + z = 24
x + 3y + 4y = 8

And they meet at line L, find the Cartesian equations for line L

Once you worked that out, you'd find your original question easy

And I know the answer because I've worked it out right now?!?
y=(8-x)/7, z=-8/7(2x-23)

I still don't find my original question easy

If you're not going to help, don't bother posting.

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JAIYEKO
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(Original post by majmuh24)
y=(8-x)/7, z=-8/7(2x-23)

I still don't find my original question easy

If you're not going to help, don't bother posting.

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Very incorrect attempt.
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(Original post by JAIYEKO)
Very incorrect attempt.
I really don't care. If you don't have anything to say with regards to the question in the thread, don't bother posting here.

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(Original post by majmuh24)
I really don't care. If you don't have anything to say with regards to the question in the thread, don't bother posting here.

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You're a GCSE student and you haven't even done and passed your GCSE Maths exam so how are you going to know AS Maths content!?

Go home
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(Original post by JAIYEKO)

You're a GCSE student and you haven't even done and passed your GCSE Maths exam so how are you going to know AS Maths content!?

Go home
By looking into it g
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(Original post by JAIYEKO)

You're a GCSE student and you haven't even done and passed your GCSE Maths exam so how are you going to know AS Maths content!?

Go home
By reading about stuff. You aware bro?

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(Original post by majmuh24)
By reading about stuff. You aware bro?

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Indeed, but you can't answer the simple question I asked you. I'm in of the finest GCSE mathematicians but I don't go making threads of AS maths.
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(Original post by JAIYEKO)
Indeed, but you can't answer the simple question I asked you. I'm in of the finest GCSE mathematicians but I don't go making threads of AS maths.
You should.

What was the answer to this simple question then?

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I have no idea. Our school laid more emphasis on solving equations even if that meant you dint have a jack clue on the reasoning. Messed up, I know.

True story!
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(Original post by majmuh24)
You should.

What was the answer to this simple question then?

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Final answer will be

Line will be (x-11)/-11 = (y+1)/-7 = z/8
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(Original post by majmuh24)
I can understand the process of differentiation and how that's used to find a tangent by using limits, but have absolutely no idea how integration gives the area under a curve Is there a similar, more intuitive explanation for integration?
I tried to explain that here: http://www.thestudentroom.co.uk/show...php?p=40024429

I know that an integral can be defined as the limit of a Riemann sum, but how do we even know that the limit of the Riemann sum converges to the integral in the first place? I think that I saw an example using the fundamental theorem of calculus, but it was pretty confusing and I don't think I understand it that well
This is a slightly odd mixture of two questions. Your first is looking for intuition, the second is talking about things you learn in a maths degree, and doesn't quite make sense.

To define an integral for some function over some range of x, say from a to b using Riemann sums, you need:

a) a finite sum of rectangles that underestimates its area, the lower sum, and
b) a finite sum of rectangles that overestimates its area, the upper sum.

You then take the limit of the upper and lower sums as you let the width of the rectangles to tend to 0, and the number of them to infinity. If both the upper and lower sums tend to the same value, then the function is said to be Riemann integrable, and the common value of the sums is the value of the integral, whose symbol is written \displaystyle \int^b_a f(x) dx.

So one way to compute the value of an integral is to evaluate the limit of upper and lower sums. However this is very tricky, usually, and you never do it.

Instead, you rely on a result called the Fundamental Theorem of Calculus, that says that if you are interested in the integral of a function f(x), and you can find some other function F(x) that differentiates to f(x), (i.e F'(x)=f(x) as with F(x)=\sin x and f(x)=\cos x) then

\displaystyle \int^b_a f(x) dx = F(b)-F(a)

To prove this, we have to show that the upper and lower sums of the Riemann integral both converge to F(b)-F(a).

So this theorem turns the process of finding the value of an integral into that of finding an anti-derivative for a function, and evaluating the value of that anti-derivative a couple of times. This is a lot easier than evaluating the limit of an infinite sum.
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It's such a shame we don't get taught this sort of stuff. If you can follow the method, most teachers are happy and wont bother explaining why we do things and wont bother with teh real maths behind teh methods.

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