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Finding the Relationship between the Limit Method and Direct Integral Method for Area

  1. Nov 30, 2011 #1
    1. The problem statement, all variables and given/known data
    I am in the process of studying integration and finding the areas under curves. So far, I know of two methods of finding the area under a curve: the limit method and the direct integral method. Could someone explain the relationship between these two methods?


    2. Relevant equations
    [itex]\int[/itex]f(x) dx = F(x)|[itex]^{b}_{a}[/itex] = F(b) - F(a) = Area

    [itex]lim_{n→∞}[/itex] [itex]\sum^{n}_{i = 1}[/itex] [itex]f(x_{i})[/itex]Δx = Area

    3. The attempt at a solution
    I noticed in the direct integration method for finding the area under a curve that the area under the curve is equal to the change in y of a more complicated function: the integral. I graphed it out on my calculator and I don't see exactly how this works.

    [itex]lim_{n→∞}[/itex] [itex]\sum^{n}_{i = 1}[/itex] [itex]f(x_{i})[/itex]Δx = Δy of F(x) = Area

    I'm trying to seek an explanation as to why the limit method yields the same result as the direct integral method.
     
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  3. Nov 30, 2011 #2

    LCKurtz

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  4. Dec 3, 2011 #3
    Re: Finding the Relationship between the Limit Method and Direct Integral Method for

    Ah yes, I read through a bit of it. I'm rather confused on the proof for the First Fundamental Theorem of Calculus where it is
    F(x) = [itex]\int^{x}_{a}f(t) dt[/itex]
    I've just never seen an antiderivative represented in this way before. Could you interpret this for me? Why does an antiderivative have an upper and lower bound?
     
    Last edited: Dec 3, 2011
  5. Dec 3, 2011 #4

    LCKurtz

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    Re: Finding the Relationship between the Limit Method and Direct Integral Method for

    Let's say you have a function f(x) and its antiderivative F(x) so you might have written [tex]F(x)=\int f(x)\, dx + C[/tex] where F'(x) = f(x). If you were going a definite integral you would write [tex]\int_a^b f(x)\,dx = (F(x)+C)|_a^b = F(b) - F(a)[/tex] and the C is usually omitted since it cancels out anyway.

    Now the x in that definite integral is a dummy variable, not affecting the answer, so that line could as well have been written[tex]\int_a^b f(t)\,dt = F(t)|_a^b = F(b) - F(a)[/tex] Since this is true for any a and b, let's choose to let b be a variable x:[tex]\int_a^x f(t)\,dt = F(t)|_a^x = F(x) - F(a)[/tex] Since these are equal you still have F'(x) = f(x) so the left side is an antiderivative of f(x). Since a can be anything, the F(a) is like the constant of integration in our first equation.

    Does that help answer your question?
     
  6. Dec 3, 2011 #5
    Re: Finding the Relationship between the Limit Method and Direct Integral Method for

    Yes!!! So, to make sure I have this correct, F(a) = C in this case, correct?
     
  7. Dec 3, 2011 #6

    LCKurtz

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    Re: Finding the Relationship between the Limit Method and Direct Integral Method for

    I would leave it as F(a). Here's an example. Suppose you are trying to find the function whose derivative is x2 and whose value at x = 0 is 4. You might do it this way:[tex]f(x) = \int x^2\, dx =\frac {x^3}{3}+C[/tex] Then you plug in x = 0 to require that f(0) = 4 and that tells you that C = 4 so your answer is[tex]f(x) = \frac{x^3} 3+4[/tex] Alternatively you could have solved the problem this way:[tex]f(x)-f(0)=\int_0^x t^2\, dt = \frac{t^3} 3 |_0^x =\frac {x^3} 3[/tex] where f(0) = 4, which you could have put in in the first place. Same answer, slightly different methods.
     
  8. Dec 3, 2011 #7
    Re: Finding the Relationship between the Limit Method and Direct Integral Method for

    If f(0) = 4, then shouldn't f(x) - f(0) = [itex]\frac{x^{3}}{3}[/itex] - 4?
     
  9. Dec 3, 2011 #8

    LCKurtz

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    Re: Finding the Relationship between the Limit Method and Direct Integral Method for

    f(x) - 4 = [itex]\frac{x^3} 3[/itex]
     
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