Calculating the integral of ((-x^2)/3)-2, from 0 to 2, using the limit definition

[sktsasus]

"f(x) = ((-x^2)/3)-2

We are supposed to calculate the integral from 0 to 2 of f(x) using the limit definition of an integral.

a) Calculate Rn for f(x), which is the Riemann sum where the sample points are chosen to be right-hand endpoints of each sub-interval on the interval [0,2]. Write your answer as a function of n without summation signs.

b) Calculate the limit of Rn as n approaches infinity."

I think the main confusion I am having is with part a. How exactly am I supposed to find the Riemann sum without knowing the length of each sub-interval? Am I supposed to make each sub-interval 1/n and then keep going until 2n/n. As these would be the right endpoints? If so, how would I calculate the sum of these seemingly infinite amount of intervals? Or am I supposed to approach this problem in a different way?

Could I also use the identity that the sum of k^2 from k = 1 to n is equal to (n(n+1)(2n+1))/6? I so, how?

Any help?

 

[Dr.Peterson]

"f(x) = ((-x^2)/3)-2

We are supposed to calculate the integral from 0 to 2 of f(x) using the limit definition of an integral.

a) Calculate Rn for f(x), which is the Riemann sum where the sample points are chosen to be right-hand endpoints of each sub-interval on the interval [0,2]. Write your answer as a function of n without summation signs.

b) Calculate the limit of Rn as n approaches infinity."

I think the main confusion I am having is with part a. How exactly am I supposed to find the Riemann sum without knowing the length of each sub-interval? Am I supposed to make each sub-interval 1/n and then keep going until 2n/n. As these would be the right endpoints? If so, how would I calculate the sum of these seemingly infinite amount of intervals? Or am I supposed to approach this problem in a different way?

Could I also use the identity that the sum of k^2 from k = 1 to n is equal to (n(n+1)(2n+1))/6? I so, how?

Any help?

Check the definitions or examples you have been given. Although a Riemann sum does not have to have sub-intervals of equal length, that is often assumed. So the right endpoints of the sub-intervals would be 2/n, 2*2/n, ..., up to n*2/n = 2. (You apparently forgot the interval is [0,2], not [0,1].)

What you end up with will be a finite series (for a given value of n); you can hope that it is one you know how to sum. In fact, the formula you show will be of use. (You won't know that until you write the series!)