Question

Hello esteemed math gurus :D

Hope you don't mind helping me with a quick calculus question!

The limit definition of an area is defined as such:
limit as n approaches infinity of the sum of f(c)*deltax where deltax = (b-a)/n

My question is: how do you choose a variable expression for c?

Apologies for my inability to use Latex :P I'll try to rewrite it more clearly!

Answer 1

\[\Large \lim_{n \rightarrow \infty} \sum_{i=1}^{n} f(c_i) \Delta x\]

This is what I mean. How do you come up with an expression for c sub i? When you want to find the area under a curve for a given expression f(x).

Answer 2

c sub i is any whole number from 1 to n for example 2, 3, 4, or even 5 up to n

Answer 3

c sub i is just a point in each sub interval.

Answer 4

\[c_i\] is ANY

Answer 5

number in the interval

Answer 6

although for ease of calculation if you are computing a riemann sum by hand one usually chooses either left hand endpoints or right hand endpoints

Answer 7

Ah yes - how do you set up c_i so that it represents either the left or right hand endpoint?

Answer 8

once you have your partition you know what they are from that

Answer 9

say you make
\[\Delta x=\frac{b-a}{n}\] and say you want left hand end points, starting at \(a\)
then
\[c_0=a, c_1=a+\Delta x, c_2=a+2\Delta x, ...,c_k=a+(k-1)\Delta x\]

Answer 10

oops should have been
\[c_k=a+k\Delta x\]

Answer 11

Ah - and "a" of course depends on what your leftmost point is? Is k the same variable as the "index of summation" then?

Answer 12

Thanks all for your replies :)

Answer 13

yes on both