integrate f(x) from 0 to n. where n is a positive integer and f(x) is the greatest integer function.

I Know that i have to use the definiton of greatest integer function and break it into sum of the integrals how do i do this.

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- Dec 7th 2006, 08:27 AMmyoplex11integration and the greatest function help exam in 7 hrs
integrate f(x) from 0 to n. where n is a positive integer and f(x) is the greatest integer function.

I Know that i have to use the definiton of greatest integer function and break it into sum of the integrals how do i do this. - Dec 7th 2006, 08:32 AMAfterShock
- Dec 7th 2006, 08:42 AMmyoplex11
this is the problem we had on our practice exam

- Dec 7th 2006, 08:51 AMThePerfectHacker
Okay.

A defined function on a closed interval is Riemann integrable if and only if it is continous almost everywhere.

We we need to find,

$\displaystyle \int_0^n [ x ] dx$

Since it is continous almost everywhere we can use the subdivision rule,

$\displaystyle \sum_{k=1}^n \int_{k-1}^k [x] dx$

But,

$\displaystyle \int_{k-1}^k [x] dx =k-1$

Because in that $\displaystyle [x]=k-1$ on this interval.

Thus,

$\displaystyle \sum_{k=1}^n k-1=\frac{n(n-1)}{2}$ - Dec 7th 2006, 08:52 AMThePerfectHacker
- Dec 7th 2006, 08:52 AMPlato
If you draw a graph for say n=8, you will see a series of seven ‘stair steps’.

The area under the ‘stairs’ is the sum of the areas of seven rectangles.

So 1+2+3…+7. To do it for n we get a sum of $\displaystyle \frac{{\left( {n - 1} \right)n}}{2}.$