Identifying Riemann sums Fill in the blanks with an interval a...

Question

Identifying Riemann sums Fill in the blanks with an interval and a value of n.

4

∑ ƒ (1 + k) • 1 is a right Riemann sum for f on the interval [ ___ , ___ ] with

k = 1

n = ________ .

Accepted Answer

Hello. In this video, we are given the Riemann sum, which we are told is using right Rieman sums for F, and we want to determine which intervals that this sum lies on, and how many sub-intervals we're using for the given sum. Now, the summation given to us is the summation from K is equal to 1 to 6 of F of 3 + 2k multiplied by 2. Now, in order to approach this problem, we are going to use the definition of the finite or of the definite integral. That definition is given as the following. The definite integral from A to B of F of XD X is equivalent to the summation from I is equal to 1 ton of F of Xi multiplied by a change in X, which is delta X. Here, delta X is defined as B minus A divided by N, and our XI is defined as A plus I multiplied by delta X. So, in order to approach this problem, let's go ahead and first identify how many sub-intervals we are working with. That is going to be the value of N. Now, N in the given definition is the stopping point of the summation, and if we take a look at the given summation, that stopping point is given to us as 6. So that tells us that the amount of subintervals that we are working with is going to be 6 for the summation. Now that we have the amount of intervals, what we want to do next is we want to go ahead and define the total interval we're working over. Let's go ahead and start by defining our delta X. Now, changing our our X I in this case. Now, X sub I is going to be the variable inside of the function in the summation, and that is given to us as 3 + 2k. So we know that Xi is equal to 3 + 2k. Now, taking a look at this, notice that the first term is 3 in the sum. That is going to be our lower interval, which is A. So that means that A for the interval is going to equal to 3. And now what we need to do is we need to solve for our B term, which is going to be the upper interval. Now, we can solve for this by using our delta X function. Delta X is the value that we're multiplying F of X of I by, and in this case here, that value is given to us as 2. So we know that Delta X is equal to 2. We also know the value of A, which is our lower boundary, and we know the value of our sub-intervals. So by using the definition of delta X, we can write that delta X is equal to B minus A divided by N. We know the value of delta X is 2, so we can substitute that to the left-hand side of this equation. We don't know our upper boundary B, so we're gonna leave that as a variable. Now, we're going to subtract our lower boundary A, which is 3, and divide by the sub-intervals, which is 6. And what we need to do now is solve this equation in order to get our variable B. Multiplying both sides by 6 will give us 12 is equal to B minus 3, and adding 3 to both sides is going to give us B is equal to 15. This is going to be the value of the upper limit for our bounds, and that tells us that the total interval we're working over is going to be from 3 to 15. So just to recap, we just solved that the total interval that the sum is over is from 3 to 15, and the amount of sub-intervals we're working with is N is equal to 6. And with that being said, the overall solution to this problem is going to be B. So I hope this video helps you in understanding how to approach this problem, and we will go ahead and see you all in the next video.

Identifying Riemann sums Fill in the blanks with an interval and a value of n.

4
∑ ƒ (1 + k) • 1 is a right Riemann sum for f on the interval [ _ , _ ] with
k = 1
n = ________ .

Key Concepts

Riemann Sums

Definite Integral

Interval Notation

Watch next