Finding exact area with the Definite Integral

**janvdl** · November 14th 2007, 11:08 PM

Okay i was busy trying to figure these things out and then i got stuck.

Now using the Riemann notation to calculate the area under the graph $x^2 + 1$ we get the following:

$R_{6} = \sum_{i = 1}^{6} [ f(x_{i}) \cdot ( \frac{3-0}{6})]$

(I calculated $R_{6} = 14,375$ )

They use rectangles under the graph to approximate the area. (6 rectangles and the x-axis value ranges from 0 to 3.)

Then the book goes further and says:

For any number of rectangles under the graph(Still ranging from 0 to 3 on the x-axis):

$R_{n} = \sum_{i = 1}^{n} [ f(x_{i}) \cdot ( \frac{3-0}{n})]$

At the end we arrive at the following:

$R_{n} = \frac{27}{n^3} \sum_{i = 1}^{n} i^2 + \frac{3}{n} \sum_{i = 1}^{n} 1$

Up to there i understand perfectly.

Then from that we get

$R_{n} = \frac{27}{n^3} \left( \frac{ n(n+1)(2n+1) }{6} \right) + \frac{3}{n} (n)$

EDIT: posted by accident. (Would appreciate it if a mod would delete this)

**Jhevon** · November 14th 2007, 11:14 PM

Originally Posted by janvdl

Okay i was busy trying to figure these things out and then i got stuck.

Now using the Riemann notation to calculate the area under the graph $x^2 + 1$ we get the following:

$R_{6} = \sum_{i = 1}^{6} [ f(x_{i}) \cdot ( \frac{3-0}{6})]$

(I calculated $R_{6} = 14,375$ )

They use rectangles under the graph to approximate the area. (6 rectangles and the x-axis value ranges from 0 to 3.)

Then the book goes further and says:

For any number of rectangles under the graph(Still ranging from 0 to 3 on the x-axis):

$R_{n} = \sum_{i = 1}^{n} [ f(x_{i}) \cdot ( \frac{3-0}{n})]$

At the end we arrive at the following:

$R_{n} = \frac{27}{n^3} \sum_{i = 1}^{n} i^2 + \frac{3}{n} \sum_{i = 1}^{n} 1$

Up to there i understand perfectly.

Then from that we get

$R_{n} = \frac{27}{n^3} \left( \frac{ n(n+1)(2n+1) }{6} \right( + \frac{3}{n} (n)$

what is your question? how thy changed the sums to those formulas? those are identities. see the identities section here

it should not be hard to find the proofs of these identities floating around the internet if you're interested

Thread: Finding exact area with the Definite Integral

LinkBack

Thread Tools

Display

Finding exact area with the Definite Integral

Similar Math Help Forum Discussions

[SOLVED] definite integral that gives area of a region

finding the exact value of an integral which contains log and e

Finding a definite integral from other given definite integrals.

Area from a definite integral

Finding exact area with the Definite Integral

Search Tags