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Math Help - Why does integration give you area?

  1. #1
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    Why does integration give you area?

    Why does integrating a function within a limit of a to b give you the area of the graph from a to b?
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    MHF Contributor Bruno J.'s Avatar
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    There are various ways to construct the integral. Typically the integral of a continuous function f:[a,b] \rightarrow \mathbb{R} is constructed as the limit of a sequence of Riemann sums.
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    Quote Originally Posted by chengbin View Post
    Why does integrating a function within a limit of a to b give you the area of the graph from a to b?

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    No one in Particular VonNemo19's Avatar
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    Quote Originally Posted by chengbin View Post
    Why does integrating a function within a limit of a to b give you the area of the graph from a to b?
    \int_a^bf(x)dx can also be interpreted as \lim_{n\to\infty}\sum_{i=1}^nf(x_i)\cdot\Delta_ix where f(x_i) represents the height of f at some value x_i on the closed interval [a,b], with x_1=a and x_n=b. \Delta_ix=x_i-x_{i-1} represents each distinct subinterval on [a,b] (which can be interpreted as width).

    Now, with the technical stuff out of the way, you can see that \lim_{n\to\infty}\sum_{i=1}^nf(x_i)\cdot\Delta_ix is just the sum of the area of a bunch of rectangles as the width of these rectangles tends to zero.

    A=w\cdot{h}, in the riemann sum, you can see that w=\Delta_ix, and height is f(x_i).

    The sum of the area of a bunch of rectangles is an area.

    the \int symbol has stretched out over the years since newton's day but it used to look more like an S. stading for sum.

    NOTE. I choose a lack of precision here in an attempt to explain the definition of the riemann integral in an intuitive way. The precise definition is

    f continuous: if

    |\sum_{i=1}^nf(\xi_i)\cdot\Delta_ix-A|<\epsilon whenever ||\Delta||<\delta.

    then we say that A=\int_a^bf(x)dx

    \xi_i being any number on each \Delta_ix, and ||\Delta|| being the largest subinterval in the partitioned subdivision from a to b.
    Last edited by VonNemo19; July 2nd 2009 at 09:01 AM.
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  5. #5
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    Quote Originally Posted by chengbin View Post
    Why does integrating a function within a limit of a to b give you the area of the graph from a to b?
    Look at how you have defined the definite integral.

    CB
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    Quote Originally Posted by CaptainBlack View Post
    Look at how you have defined the definite integral.

    CB
    I'm sorry, I don't quite get that. Do you mind elaborating?
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    Quote Originally Posted by chengbin View Post
    I'm sorry, I don't quite get that. Do you mind elaborating?
    What definition of an integral are you using?

    CB
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    Quote Originally Posted by chengbin View Post
    I'm sorry, I don't quite get that. Do you mind elaborating?
    Then I guess we have to ask, exactly what definition of the integral you are using?
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  9. #9
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    Quote Originally Posted by chengbin View Post
    Why does integrating a function within a limit of a to b give you the area of the graph from a to b?
    Get The Calculus by Leithold and study chapter 6. Go through that chapter carefully or even just bits and pieces, and I bet you forever understand what's area got to do with definite integrals.
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    Quote Originally Posted by HallsofIvy View Post
    Then I guess we have to ask, exactly what definition of the integral you are using?
    For example if I integrate this function

    \int_0^2 x^2dx

    Integrating that will give you the area under the function y=x^2 from 0 to 2.

    Why?

    @shawsend

    I can't get that. I'm not a university student, just a 14 year old kid that finished the calculus part of Kumon math curriculum.
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    No one in Particular VonNemo19's Avatar
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    Quote Originally Posted by chengbin View Post
    For example if I integrate this function

    \int_0^2 x^2dx

    Integrating that will give you the area under the function y=x^2 from 0 to 2.

    Why?
    Did you not understand my previous post?
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  12. #12
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    Quote Originally Posted by chengbin View Post
    For example if I integrate this function

    \int_0^2 x^2dx

    Integrating that will give you the area under the function y=x^2 from 0 to 2.

    Why?

    @shawsend

    I can't get that. I'm not a university student, just a 14 year old kid that finished the calculus part of Kumon math curriculum.
    Ok sorry.
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  13. #13
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by chengbin View Post
    I can't get that. I'm not a university student, just a 14 year old kid that finished the calculus part of Kumon math curriculum.
    is it illegal for 14 year olds to get calculus books where you're from? or are you saying you don't have the money?
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  14. #14
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    Smile

    hi
    what would you do,if you're told to find the colored area in this example ?
    Attached Thumbnails Attached Thumbnails Why does integration give you area?-graph.png  
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  15. #15
    No one in Particular VonNemo19's Avatar
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    I can't get that. I'm not a university student, just a 14 year old kid that finished the calculus part of Kumon math curriculum
    I did not know that. Well, then. It is very good that you have come to where you are at such a young age. When you are my age you will have achieved much in the realm of mathematics.

    Maybe, if I say to you that, the way in which books are set up tend to favor practical use over conceptual understanding. Math books today are designed to provide engineering students the tools necessary to perform calculations. This means that the authors don' really care if you know why a theorem is what it is, they just want you to be able to go out and apply already proven theorems to real life problems. It will not be until much, much later, if you decide to major in math, that you will enter into the world of whys, and finally escape the drab world of how tos.

    If you have trouble with my previous post, and you would still like to know WHY the integral is sometimes interpreted as an area under the curve of the integrand, then you must venture out and resarch it. Continue to ask your questions, because this is a noble enterprise. One day, you will be a man. The road to in depth understanding of math is long and lonely. You will one of the few singularities of peculiararity that will wonder at the language of nature, whilst you marvel at the rabble, thinking to yourself, "Don't they ever wonder WHY?"

    Best of luck kid.
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