## byerskm2 3 years ago Find a function f for which the limit below represents the area under f on the interval [0, a]. Here we assume the rectangles used have equal widths and right endpoints as sample points. http://i49.tinypic.com/20ua7i9.jpg

1. satellite73

any guesses for this one?

2. satellite73

no guesses? i guess $$f(x)=x^2$$

3. satellite73

on the interval $$[0,1]$$ divide the interval in to $$n$$ parts each of length $$\frac{1}{n}$$

4. byerskm2

Just x^2? I thought it had to be more than that though

5. satellite73

the first simple point will be $$\frac{1}{n}$$ the second will be $$\frac{2}{n}$$ and the $$i$$th will be $$\frac{i}{n}$$

6. satellite73

then for each $$i$$, $$f(\frac{i}{n})=\frac{i^2}{n^2}$$

7. byerskm2

oh wow! Dang. I hate this notation. It makes it ten times more complicated than it needs to be.

8. satellite73

multiply each by $$\frac{1}{n}$$ and add the up you get $\sum\frac{i^2}{n^3}$

9. satellite73

well don't be hoodwinked in to thinking these are all that easy usually it is much harder to tell

10. byerskm2

I have another one a lot like this. with a sqaure root. but I'm not sure about it

11. satellite73

but the first clue should have been the $$i^2$$ on the other hand if the interval had be $$[1,2]$$ it would have been completely different

12. byerskm2

interesting, because the i and n values would be different right?

13. byerskm2

because a-b / n ?