# Summation - Riemann Intergral -

• Mattofix
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.
Mattofix
[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})

## 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...

...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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