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

**Yogi_Bear_79** · August 27th 2006, 12:17 PM

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 → ∞.

**galactus** · August 27th 2006, 02:33 PM

${\Delta}x=\frac{3-1}{n}=\frac{2}{n}$

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.

Thread: Help find the exact area A of region under f(x)

LinkBack

Thread Tools

Display

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

Similar Math Help Forum Discussions

Find the area of the region

Find the area of the region

Find area of region

find the area of the region

Please Help: find the area of the region

Search Tags