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Math Help - Help find the exact area A of region under f(x)

  1. #1
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    Help find the exact area A of region under f(x)

    Given f(x)=x^2+3, find the exact area A of the region under f(x) on the interval [1, 3] by first computing \sum_{i=1}^{n}f(x_i)\triangle x and then taking the limit as n → ∞.
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  2. #2
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    {\Delta}x=\frac{3-1}{n}=\frac{2}{n}

    Right endpoint method:

    x_{k}=a+k{\Delta}x=1+\frac{2k}{n}

    ((x_{k})^{2}+3){\Delta}x =\left(\frac{4k^{2}}{n^{2}}+\frac{4k}{n}+4\right)(  \frac{2}{n})

    =\frac{8(k^{2}+kn+n^{2})}{n^{3}}

    \frac{8}{n^{3}}\left[\sum_{k=1}^{n}k^{2}+n\sum_{k=1}^{n}k+n^{2}\sum_{k=  1}^{n}{1}\right]

    \frac{8}{n^{3}}\left[\frac{n(n+1)(2n+1)}{6}+\frac{n^{2}(n+1)}{2}+{n^{3}  }\right]

    \lim_{n\to\infty}\left[\frac{8}{n}+\frac{4}{3n^{2}}+44/3\right]

    Now, see what the limit is?. If you simply integrate x^{2}+3 from 1 to 3 you'll get the same result.
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