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Integral of a area under a straight line as summation

  1. Apr 21, 2017 #1
    1. The problem statement, all variables and given/known data
    Snap2.jpg

    2. Relevant equations
    Summs:
    $$1+2+3+...+n=\frac{n(n+1)}{2}$$
    $$1^2+2^2+3^2+...+n^2=\frac{n(n+1)(2n+1)}{6}$$

    3. The attempt at a solution
    $$\Delta x=\frac{b}{n}$$
    $$S_n=f\left( \frac{\Delta x}{2} \right)\Delta x+f\left( \Delta x+\frac{\Delta x}{2} \right)\Delta x+...+f\left( (n-1)\Delta x+\frac{\Delta x}{2} \right)\Delta x$$
    $$S_n=\left( \frac{b}{2n} \right) \frac{b}{n}+\left( \frac{b}{n}+\frac{b}{2n} \right) \frac{b}{n}+\left( 2\frac{b}{n}+\frac{b}{2n} \right) \frac{b}{n}+...+\left( (n-1)\frac{b}{n}+\frac{b}{2n} \right) \frac{b}{n}$$
    $$S_n=\frac{b^2}{2n^2}(1+3+5+...+2n-1)$$
    But i was taught, in that chapter, only the two sums from the Relevant Equations
     
  2. jcsd
  3. Apr 22, 2017 #2

    LCKurtz

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    Here's a hint to get you a better start. Let ##\Delta x_k = x_k - x_{k-1}## and ##c_k = \frac{x_k+x_{k-1}}{2}## with ##x_0=0,~x_n = b##. Then$$
    S_n = \sum_{k=1}^n f(c_k)\Delta x_k =\sum_{k=1}^n \frac{(x_k+x_{k-1})}{2}(x_k - x_{k-1})$$Try simplifying that product in the sum and write out a few terms to see what happens. I don't think you will need any of your fancy sum formulas.
     
  4. Apr 22, 2017 #3

    vela

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    And if you want to continue with your attempt as well, consider the sum ##1+2+\cdots+(2n-1)+2n## and split it up into a sum of the even and odd terms.
     
  5. Apr 22, 2017 #4
    $$S_n=\frac{x_1+x_0}{2}(x_1-x_0)+\frac{x_2+x_1}{2}(x_2-x_1)+\frac{x_3+x_2}{2}(x_3-x_2)+...+\frac{x_n+x_{n-1}}(x_n-x_{n-1})=...=\frac{1}{2}x_n^2$$
     
  6. Apr 23, 2017 #5

    LCKurtz

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    Do you see why it equals ##\frac{1}{2}x_n^2##? Did you multiply out the products in the sum as I suggested in post #2? Do you understand how to get the answer, or are you just copying stuff?
     
  7. Apr 23, 2017 #6
    I know how to get to the answer, i just shortened. but i don't understand the other method suggested:
    I calculated that ##~1+3+5+...+n=n^2##, but not as a split sum of even and odd numbers.
     
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