Summation - Riemann Intergral -

In summary, the conversation discusses working on the upper and lower Riemann sums of the function f(x) = exp(-x), with Pn representing the partition of [0,1] into n subintervals of equal length. The upper sum is given as the sum from i=1 to n of exp(-i/n)/n and the lower sum as the sum from i=1 to n of exp((1-i)/n)/n. The next step is to take the limit as n-> infin to obtain the final upper sum.
  • #1
Mattofix
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[SOLVED] Summation - Riemann Intergral - URGENT

Homework Statement



Im working on the upper and lower riemann sums of f(x) = exp(-x)

where Pn donates the partition of [0,1] into n subintervals of equal length (so that Pn = {0,1/n,2/n,...,1})


Homework Equations





The Attempt at a Solution



So far i have the upper sum to be the sum from i=1 to n of exp(-i/n)/n - is this right? - if so where do i go from here? I think I am meant to take the limit when n-> infin which should give the final upper sum. I hope this right...
 
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  • #2
...and the lower sum to be the sum from i=1 to n of exp((1-i)/n)/n ?
 

1. What is the Riemann integral and how does it relate to summation?

The Riemann integral is a method used in calculus to calculate the area under a curve. It is closely related to summation because it involves dividing the area into smaller rectangles and finding the sum of their areas to approximate the total area.

2. What is the difference between a left Riemann sum and a right Riemann sum?

A left Riemann sum uses the left endpoint of each subinterval to calculate the height of the rectangle, while a right Riemann sum uses the right endpoint. This can result in different approximations for the area under a curve.

3. How can the Riemann integral be used to find the area under a curve?

The Riemann integral involves taking the limit of a summation of smaller rectangles as the width of the rectangles approaches zero. This limit gives the exact area under the curve.

4. Can the Riemann integral be used for non-continuous functions?

No, the Riemann integral can only be used for continuous functions. This means that the function must be defined at every point along the interval being integrated.

5. Are there any limitations to using the Riemann integral for calculating area?

Yes, the Riemann integral can only be used for functions that are integrable, meaning they have a well-defined area under the curve. Some functions, like the Dirichlet function, are not integrable and therefore cannot be calculated using the Riemann integral.

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