Can anyone help with finding the Riemann sum for f(x)=x^3?

In summary, the conversation is about finding the Riemann sum for a given function and interval, specifically for the function f(x)=x^3 on the interval [1,5]. The first step is to find the Riemann sum for an equipartition P=(1,2,3,4,5) into 4 intervals with right-hand endpoints. Then a formula for the Riemann sum for an equipartition P_n into n intervals and right-hand endpoints is given using the summation formulae sum(i^3)=1/4n^2(n+1)^2, with an upper limit of n and lower limit of i=1. The person asking for help is advised to provide their progress and
  • #1
dan
Hello there, can anyone help me here as I'm finding it difficult to tackle this question.

Consider f(x)=x^3 on the interval [1,5].
Find the Riemann sum for the equipartition P=(1,2,3,4,5) into 4 intervals with x_i^* being the right-hand endpoints (ie. x_i=a+hi)

Then find a formula for the Riemann sum for an equipartition P_n into n intervals and right-hand endpoints. Use the summation formulae
sum(i^3)=1/4n^2(n+1)^2, upper lim=n, lower lim;i=1.

Cheers
 
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  • #2
Hi dan,
I believe this belongs in Homework Help and the policy there is that you tell us what you got so far, and where you are stuck. Then we can help.
 
  • #3
for posting your question!

To find the Riemann sum for f(x)=x^3, we first need to understand the concept of Riemann sums. A Riemann sum is a method for approximating the area under a curve by dividing the interval into smaller subintervals and using the heights of these subintervals to calculate the total area. In this case, we will be using the right-hand endpoints of each subinterval to calculate the Riemann sum.

For the given function f(x)=x^3 on the interval [1,5], we can divide the interval into 4 subintervals of equal length, with the endpoints being 1, 2, 3, 4, and 5. This is known as an equipartition. The right-hand endpoints of these subintervals will be x_i=1+hi, where h is the length of each subinterval.

To calculate the Riemann sum, we use the formula: R_n=∑f(x_i*)Δx, where f(x_i*) is the height of each subinterval and Δx is the width of each subinterval.

In this case, f(x_i*)=(1+hi)^3 and Δx=1. Therefore, the Riemann sum for this equipartition is:

R_4=[(1+h)^3+(1+2h)^3+(1+3h)^3+(1+4h)^3]h

To find a formula for the Riemann sum for an equipartition P_n into n intervals and right-hand endpoints, we use the summation formula sum(i^3)=1/4n^2(n+1)^2, with the upper limit being n and the lower limit being i=1. This gives us the formula:

R_n=1/4n^2(n+1)^2*h

Where h is the width of each subinterval, which can be calculated by dividing the length of the interval by the number of subintervals (n).

I hope this helps you with your question. Remember to always double-check your calculations and make sure your Riemann sum is an approximation and not an exact value. Good luck with your studies!
 

1. What is a Riemann sum?

A Riemann sum is a method for approximating the area under a curve by dividing the region into smaller rectangles and adding up their individual areas.

2. How do you find the Riemann sum for a given function?

To find the Riemann sum for a given function, you need to first choose the number of rectangles you want to use to approximate the area. Then, you calculate the width of each rectangle by dividing the total interval by the number of rectangles. Finally, you evaluate the function at each rectangle's midpoint and multiply it by the width of the rectangle. Add all of these values together to get the Riemann sum.

3. Why is the Riemann sum important?

The Riemann sum is important because it is a fundamental concept in calculus and is used to calculate the area under a curve. It is also used to approximate the value of a definite integral, which is crucial in solving many real-world problems.

4. Can you explain the different types of Riemann sum?

There are three main types of Riemann sum: the left Riemann sum, the right Riemann sum, and the midpoint Riemann sum. These differ in the placement of the rectangles under the curve. The left Riemann sum uses the left endpoint of each interval, the right Riemann sum uses the right endpoint, and the midpoint Riemann sum uses the midpoint of each interval to evaluate the function.

5. How can I check if my Riemann sum is accurate?

You can check the accuracy of your Riemann sum by increasing the number of rectangles used to approximate the area. The more rectangles you use, the closer your approximation will be to the actual area under the curve. You can also compare your Riemann sum to the exact solution, if it is known.

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