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Math Help - approximating area, sigma sum/limit equation

  1. #1
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    approximating area, sigma sum/limit equation

    The problem
    Use Definition 2 to find an expression for the area under the graph of f as a limit. do not evaluate the limit.
    Definition 2 is

     A = \lim_{n->\infty} R_{n}= \lim_{n->\infty} [f(x_{1})\Delta x+f(x_{2})\Delta x+....+f(x_{n})\Delta x]

    17. f(x)=\sqrt[4]{x}, 1 \leq x \leq 16

    I got
    \lim_{n->\infty} \sum^n_{i=1}  [f(x_{i}) \times<br />
  \frac{b-a}{n}]<br />

    =\lim_{n->\infty} \sum^n_{i=1} [f(x_{i} \times \frac{15}{n}]

    =\lim_{n->\infty} \sum^n_{i=1} [f(\frac{15i}{n})\times\frac{15}{n}]


    =\lim_{n->\infty} \sum^n_{i=1} [\sqrt[4]{\frac{15i}{n}}\times \frac{15}{n}]

    but the book's answer is

    \lim_{n->\infty} \sum^n_{i=1} [\sqrt[4]{1 + \frac{15i}{n}}\times \frac{15}{n}]

    where did that extra 1 come from?
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  2. #2
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    That "1" is the initial x value. Using your x_n= \frac{15i}{n}, when i= 0, x_0= 0 and when i= 15, x_15= 15. But you want the integral to go from 1 to 16, not 0 to 15.

    In general, if the integral is from a to b, with n steps, you want the step to be \frac{b- a}{n} but then the x values will be a, a+ \frac{b-a}{n}, a+ \frac{2(b-a)}{n}, \cdot\cdot\cdot, a+ \frac{i(b-a)}{n}, \cdot\cdot\cdot.
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  3. #3
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    what if y=c?
    I have a problem where I have to find the integral from 2 to 3 and the points of the graphs are (2,3) and (3,3) so y=3 or f(x)=3
    when evaluating the integral, i know that there is a rule where integral of a to b =c dx =c(b-a)
    but if you didn't know that rule and just did it the long way with the reiman's sum method, would you still have to add that initial x? in this case 2?
    I tried using the reiman's sum method and adding 2 and I got the wrong answer. 5 instead of 3.
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