Here's the question you clicked on:
byerskm2
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
any guesses for this one?
no guesses? i guess \(f(x)=x^2\)
on the interval \([0,1]\) divide the interval in to \(n\) parts each of length \(\frac{1}{n}\)
Just x^2? I thought it had to be more than that though
the first simple point will be \(\frac{1}{n}\) the second will be \(\frac{2}{n}\) and the \(i\)th will be \(\frac{i}{n}\)
then for each \(i\), \(f(\frac{i}{n})=\frac{i^2}{n^2}\)
oh wow! Dang. I hate this notation. It makes it ten times more complicated than it needs to be.
multiply each by \(\frac{1}{n}\) and add the up you get \[\sum\frac{i^2}{n^3}\]
well don't be hoodwinked in to thinking these are all that easy usually it is much harder to tell
I have another one a lot like this. with a sqaure root. but I'm not sure about it
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
interesting, because the i and n values would be different right?