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Riemann Sum problem

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  1. Apr 23, 2015 #1
    1. The problem statement, all variables and given/known data

    Hello, thank you in advance for your help. I am calculating a Riemann sum with right hand endpoints. I hit a small snag, and I appreciate your help in getting me straight.


    2. Relevant equations

    f(x) = x2+ 1, over the interval [0,1]. This is problem number such-and-such from a well-known calculus textbook, not anything that is on an exam that I know of. The limit of these sums approaches infinity.

    I'll call the Riemann Sum Sp, the length (x value) of the rectangles ∆x, and the height of the rectangles (y value) ck.


    3. The attempt at a solution

    ∆x = 1/n, good to go.

    ck = a + k[(b - a)/n] = 0 + k (1/n) = k/n

    Sp = Σ(k = 1, n) k/n * 1/n + 1

    Sp = 1/n2 Σ(k = 1, n) k + 1

    And here is where I ran out of juice.

    Again, and help is appreciated.

    SY
     
  2. jcsd
  3. Apr 23, 2015 #2

    LCKurtz

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    Since ##f(x) = x^2 +1## and ##c_k = \frac k n##, then ##f(c_k) = \frac {k^2} {n^2}+1## so you should be calculating$$
    \sum_{k=1}^n(\frac {k^2} {n^2}+1)\frac 1 n$$

    You will need the formula for the sum of squares of the first n integers, which you can look up.
     
  4. Apr 23, 2015 #3
    I believe I just pulled a first-year calculus moron maneuver. Instead of k, it should be k2? And each term would then proceed to be determined using sigma notation for the degree of the individual term?

    Here, that would be (as I see it),

    Sp = 1/n2 *
    ∑k=1n k2 + ∑k=1n 1
    = 1/n2 * [n(n+1)(2n+1)/6] + (n)-- my apologies for this being superscripted, I am having an issue w/ the formatting

    Here, am I correct in understanding with the first sigma term, that reduces to n * 1/n2?

    If I am, them we have 1/n * [(2n3+ n2+ 2n + n)/6]

    = (2n3+ n2+3n)/6n

    = (2n2+n + 3)/6

    As I understand it, if i substitute the limit for n, infinity, then I am left with an area of 2/6 = 1/3 sp units. That doesn't pass the sniff test. What did I do wrong?

    Thank you for your time.
     
  5. Apr 23, 2015 #4

    SammyS

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    Read post #2.
     
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