Question

Can someone explain to me what this means?

Answer 1

Also, when working with reimman sums the point Ck is the one that determines the heigh of an intervals right, and we get that point ck from any interval in the partition?

Answer 2

It should be the product of the height of the interval and the size of the interval.

Answer 3

Ck is the middle of the kth interval of the partition. so C1 is the middle of the first interval. f(Ck) is the height corresponding to the kth interval.

Answer 4

\[\sum_{k=1}^{n}\frac{f(c_{k})}n{}\]
So this means the sum of all the heights corresponding to the intervals divided by n.

Answer 5

but for that point ck , it can either be the middle or the right or left end point right

Answer 6

Oh, yeah that's right, I was a little sloppy there.

Answer 7

which means that all of subintervals will have to evaluated at with the right left or middle endpoint right? no mater if if the subintervals have the same width

Answer 8

same width or not

Answer 9

Yes what you have there implies a partition of size one with each interval being size 1/n.

Answer 10

Look at the sum its from 1 to n with 1/n partitions.

Answer 11

so, is ck then just an arbritay point that i can pick within anyone of the subintervals

Answer 12

Yes, it doesn't matter.

Answer 13

But the real definition of the riemann sum is for arbitrary partitions.

Answer 14

right, by arbritay partitions you mean it could be an given number of subintervals and they could all vary in size right?

Answer 15

The partition could be any size, and the intervals don't have to be the same size.

Answer 16

but the point ck determins where all the other subintervals will get evaluated at correct? like if ck is a mid point, then all the other subintervals have to be evaluated at the mid point

Answer 17

It doesn't matter since no matter where you pick the point in each interval at the limit you will get the same result.

Answer 18

Right, so you can choose that per interval too.

Answer 19

so then what is th importance of ck

Answer 20

wait, so i can choose to evluate certain intervals at right enpoints, other at left endpoint, and maybe a few at mid point?

Answer 21

The importance is that you are considering a height for each interval. It needs to be related to the function on that interval.

Answer 22

Of course if you are using Riemann sums for approximation, some choices might be better than others.

Answer 23

In many cases you will be using Riemann sums as a theoretical tool. You will consider limits later and then it only matters what happens as n (the number of intervals) grows very large

Answer 24

But for the subintervals they all have to be evaluated at right endpoint, left endpoints or mid points right,

Answer 25

can someone walk me through one problem , just so i can get this down

Answer 26

You can pick any point in the interval, 1/4 of the interval if like that for example.

Answer 27

f(x)=1+sinx [-pi,pi]

Answer 28

It ask to partition the interval into four subintervals of equal length

Answer 29

Then add to you sketch the rectangles associated wit the reimman sum ( in attactment), given that ck is a)left hand endpoint, b)right hand endpoint, c)midpoint of the kth interval

Answer 30

Let's start with the intervals.

Answer 31

can we sketch the graph first??

Answer 32

http://www.wolframalpha.com/input/?i=Plot [1%2BSin[x]%2C{x%2C-pi%2Cpi}]

Answer 33

http://www.twiddla.com/575038

Answer 34

Midpoint: http://mathworld.wolfram.com/webMathematica/RiemannSum.jsp?fun=1%2BSin%5Bx%5D&xMin=-Pi&xMax=Pi&n=4&sample=5&estimate=True&rectstyle=1&plotstyle=5&nt=1
Left: http://mathworld.wolfram.com/webMathematica/RiemannSum.jsp?fun=1%2BSin%5Bx%5D&xMin=-Pi&xMax=Pi&n=4&sample=3&estimate=True&rectstyle=1&plotstyle=5&nt=1
Right: http://mathworld.wolfram.com/webMathematica/RiemannSum.jsp?fun=1%2BSin%5Bx%5D&xMin=-Pi&xMax=Pi&n=4&sample=4&estimate=True&rectstyle=1&plotstyle=5&nt=1

Answer 35

can we still do the problem, so that i can see the process, if its not too much of an incovenience

Answer 36

right

Answer 37

http://www.twiddla.com/575046