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Compute ∫√(25 - x^2) dx from 0 to 5 using an infinite Riemann Sum

  1. Sep 28, 2014 #1

    s3a

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    1. The problem statement, all variables and given/known data
    Integrate √(25 - x^2) dx from 0 to 5 using an infinite Riemann Sum

    2. Relevant equations
    lim n→∞ Σ_(i=1)^n i = n(n+1)/2
    lim n→∞ Σ_(i=1)^n i^2 = n(n+1)(2n+1)/6

    3. The attempt at a solution
    Δx = (b - a)/n
    Δx = (5 - 0)/n
    Δx = 5/n

    f(x_i) = √(25 - [a + iΔx]^2)
    f(x_i) = √(25 - [0 + 5i/n]^2)
    f(x_i) = √(25 - [5i/n]^2)
    f(x_i) = √(25 - 25 i^2/n^2)
    f(x_i) = √(25) √(1 - i^2/n^2)
    f(x_i) = 5 √(1 - i^2/n^2)

    lim n→∞ Σ_(i=1)^n [f(x_i) Δx]
    lim n→∞ Σ_(i=1)^n [ [5 √(1 - i^2/n^2)] [5/n] ]
    lim n→∞ 5/n Σ_(i=1)^n [5 √(1 - i^2/n^2)] (This is where I'm stuck.)

    Is it impossible to compute the definite integral of √(25 - x^2) dx from 0 to 5 using an infinite Riemann sum (such that I have to use the regular integral method of trigonometric substitution instead)?

    If it is possible, how do I proceed from where I am stuck?

    Any help in getting unstuck would be GREATLY appreciated!
     
  2. jcsd
  3. Sep 29, 2014 #2
    Your work looks correct, but I am unaware of any way to compute ##\lim\limits_{n\rightarrow\infty}\sum\limits_{i=1}^n5\sqrt{1-\frac{i^2}{n^2}}\frac{5}{n}## without identifying it as a definite integral (which we already know here) and doing it the "easy" way. For the record the "easy" way involves knowing what the graph of ##y=\sqrt{25-x^2}## is along with some very basic geometry.

    Also, I'm not a fan of the terminology "infinite Riemann sum". Riemann sums are finite. Integrals are defined to be limits of Riemann sums. There are no infinite Riemann sums.
     
  4. Sep 29, 2014 #3

    pasmith

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    Homework Helper

    One method is, in effect, to prove a special case of the fundamental theorem by finding an antiderivative [itex]F: [0,5] \to \mathbb{R}[/itex] of [itex]\sqrt{25 - x^2}[/itex] and applying the mean value theorem to [itex]F[/itex] on each subinterval of an arbitrary partition to conclude that [tex]
    \int_0^5 \sqrt{25 - x^2}\,dx = F(5) - F(0).[/tex]
     
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