Find a formula for the Riemann sum and take the limit of the sum as n->infinite

In summary, the problem given is to find a formula for the Riemann sum obtained by dividing the interval [1,5] into n equal subintervals and using the right-hand endpoint for each c subscript k. Then, taking the limit of this sum as n approaches infinity will calculate the area under the curve over [1,5]. The poster, tornado711, was able to guess the solution for part a and is now seeking help with parts b and c. However, it is not clear what parts b and c refer to and the forum rules state that users must show their work and clarify their question.
  • #1
tornado711
1
0
For the function given below find a formula for the Riemann sum obtained by dividing the interval [1,5] into n equal subintervals and using the right-hand endpoint for each c subscript k. Then take a limit of thissum as n-> infinite to calculate the area under the curve over [1,5].

Below you can see the problem. I was able to guess them correctly, so that I could perhaps try to work backwards to get it. Below is the problem. I found a. but i need to find parts b and c how do I do this?

Thanks,
MattView attachment 5501
 

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  • #2
Hi tornado711 and welcome to MHB!

It's not clear (at least to me) what you mean by "parts b and c". I don't see a reference to them anywhere on the attached image. Also, we ask (and our forum rules state) that users show their work, or any ideas on where to begin, when posting a problem. This eliminates the possibility of a redundant reply and gives us a clear idea of exactly what it is that you need help with. Can you post what you have done so far and clarify what you mean by "parts b and c"?
 

What is a Riemann sum?

A Riemann sum is a mathematical concept used to approximate the area under a curve by dividing the region into smaller rectangles and summing up their areas. It is an integral calculus technique and is used to find the exact value of an integral when the function is not easily integrable.

How do you find a formula for a Riemann sum?

The formula for a Riemann sum is ∑f(x*)Δx, where f(x*) is the value of the function at a selected point within each subinterval and Δx is the width of each subinterval. This formula can be used to find an approximation of the area under a curve by dividing the region into smaller rectangles.

What is the limit of a Riemann sum as n approaches infinity?

The limit of a Riemann sum as n approaches infinity is equal to the definite integral of the function over the given interval. This means that as the number of subintervals increases, the approximation of the Riemann sum gets closer to the exact value of the integral.

How do you take the limit of a Riemann sum as n approaches infinity?

To take the limit of a Riemann sum as n approaches infinity, we can use the limit notation limₙ→∞ ∑f(x*)Δx. This means that we are finding the limit of the sum as the number of subintervals, n, approaches infinity. We can also use the integral notation ∫f(x)dx to represent the limit of the Riemann sum as n approaches infinity.

Why is taking the limit of a Riemann sum as n approaches infinity important?

Taking the limit of a Riemann sum as n approaches infinity is important because it allows us to find the exact value of an integral even when the function is not easily integrable. It also helps us to improve the accuracy of our approximation of the area under a curve by using smaller and smaller subintervals. This concept is essential in many areas of mathematics and physics, such as in finding the area under a velocity-time graph to determine the displacement of an object.

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