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Conceptual question of integration and derivation

  1. Dec 21, 2009 #1
    Okay I have several questions that have been bothering me for some time now, so bear with me please.

    • How does integration work? Integrating the function f(x) between the bounds a,b finds the area under f(x) between a and b. But how does this work?


      If my function is f(x) = 2x and my bounds are [1,10], how come F(10) - F(1) equals the area under 2x from 1,10? (Where F(x) = x^2).

    • Furthermore, why does a derivative work? The derivative of the function f(x) yields a relationship between the rate of change in f(x) and a given input. But how does this actually work?

      Or put in a different way, if my function is 1x^2, why can I multiply the coefficient by the exponent to yield a function (2x) that tells me the rate at which the function will change?

    • What does it mean that time is being removed (divided in a derivative) or added (multiplied in an integral)?

    Thanks for reading.
     
  2. jcsd
  3. Dec 22, 2009 #2
    Okay, God can tell you how it work. Human just try to compute and believe in conformation.
     
  4. Dec 22, 2009 #3
  5. Dec 22, 2009 #4
    The best way to think of why it works is as follows:
    the average rate of change on f(x) between x and x+h where h is some constant will be (f(x+h)-f(x))/h If we take the limit as h->0 for this expression then this calculates the instantaneous rate of change, or the derivative at x, since we are finding the rate of change between x and x+0, or x. The best way to visualise this is as a straight line between two points, A=(x,f(x)) and B=(x+h,f(x+h)), and then imagine B coming along the line closer and closer to A, the line will aproach the tangent at point A, which is its rate of change at that moment.

    For integration the same sort of idea can be used, this time imagine large rectangles approximating the area under a curve. As the width of these rectangles is lowered and a larger number of rectangles are used, the approximate area under the curve will become more accurate to the real area. If we take the limit as width->0 of the sum of the area of all these rectangles then you will get the area under the curve.
     
  6. Dec 22, 2009 #5

    Landau

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    That's an insult to mathematicians. Or are you saying mathematicians are not human?

    On topic: these are questions that span a large portion of calculus, why don't you just take a look at a (good) calculus book (with proofs) and then ask more specific questions?
     
  7. Dec 22, 2009 #6

    HallsofIvy

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    Well, there are people, called mathematicians, who assit God in this work.
     
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