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Calculus - strange anomaly? can anyone explain

  1. Jul 23, 2003 #1
    f(x) = (1 - x^2)^1/2

    This all stems from me approximating pi by numerically evaluating the integral S f(x)dx from 0 to 1 and multiply the sum by 4.


    Would you agree that f(x) has a derivative
    f'(x) = (1 - x^2)^-1/2 * -2x


    According to my textbook this is so. Now I can easily find a primary function for f(x).

    F(x) = (1 - x^2)^3/2 / -2x

    Now it doesn't seem possible to evaluate [ F(x) ] from 0 to 1.
    Though it should yeild pi/4, it doesn't.

    Doing a riemann sum produces an approximation to pi, while evaluating [ F(x) ] only returns bogus. Since pi is an irrational number I accept that it is impossible to express it exactly. Though, I would like someone to explain why this doesn't work.
  2. jcsd
  3. Jul 23, 2003 #2
    If you plot the function

    it is of course a circle centered on x and y equals zero, and has an infinite derivative at x=&plusminus;1.
  4. Jul 23, 2003 #3
    Your F(x) is not the antiderivative to f(x) = sqrt(1-x2). It should be F(x) = (1/2)x sqrt(1-x2)+arcsin(x)/2. This new F(x) evaluates the function properly.
  5. Jul 23, 2003 #4
    Ok thanks alot! Something must be left out of my textbook in that case.

    Can you give me more info on this?
    Last edited: Jul 23, 2003
  6. Jul 23, 2003 #5


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    Try differentiating F(x); you'll see that it does not come out to f(x).

    As for evaluating the integral, look through the section on trigonometric substitution.
  7. Jul 24, 2003 #6


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    Does your textbook have the "fundamental theorem of calculus"?

    If you agree that the derivative of f= (1 - x^2)^1/2 is
    f'(x) = (1 - x^2)^-1/2 * -2x then obviously a "primary" function for f' is f itself, not the formula you give.
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