Left and right Riemann sums Complete the following steps for t...

Question

Left and right Riemann sums Complete the following steps for the given function, interval, and value of n.

f(x) = x + 1 on [0,4]; n = 4

(d) Calculate the left and right Riemann sums.

Accepted Answer

Hello. In this video, we are going to be finding the right rhyme and sums for the function F of X is equal to 3X minus 2 on the interval from 2 to 6 with N is equal to 4 subintervals. So, in order to find the right rhyme and sum, the first thing we want to do is we want to go ahead and define the end points for the interval from 2 to 6. Let's go ahead and make a number line from 2 to 6. Now, we are going to work with 4 sub-intervals, and so we're going to use this and the overall interval, and first we are going to define the width of each of the sub-intervals. That width is going to be defined as delta X, and delta X is equal to the larger boundary minus the smaller boundary, divided by the amount of sub-intervals we are working with N. Now, in this case here, are boundaries from 2 to 6. So, Delta X is going to be defined as 6 minus 2 divided by 4. This will be simplified to 4 divided by 4, which is going to have a value of 1. And so what this means is that each of our intervals is going to have a width of 1. So again, we are dividing this number line into 4 sub-intervals, and we are going to add 1 until we go from 2 to 6. So 2 + 1 is 33 + 1 is 44 + 1 is 5, and 5 + 1 is 6. So, these are going to be the intervals or the end points for the sub-intervals. Now, what's important here is that we want to find the right rhyme and sum for the function. Now, since we are working with the right Riemann sum, we want to use the 1st 4 right endpoints. Those endpoints are going to be 345, and 6. And so now what we want to do is we want to find the value of the function at these endpoints. If we take the first endpoint and plug it into the function, we get 3 multiplied by 3 minus 2. This is going to give us the value of 7. Plugging in 4 into the function, gives us 3 multiplied by 4 minus 2, and that gives us the value of 10. Plugging in 5 into the function gives us 3 multiplied by 5 minus 2, and this is going to simplify to be 13. And finally, plugging in 6 into the function is going to give us 3 multiplied by 6 minus 2, and that is going to give us the value of 16. So now that we have the value of the function at our right endpoints, let's go ahead and create the right rhyme and sum for the function. That right Ryman sum is going to be defined as the summation from 1 to 4 of F of X sub I multiplied by delta X. This is going to further expand to be delta X multiplied by F of 3. Plus F of 4 plus F of 5 plus F of 6. And since we just solved for the values of the function and delta X, this summation is going to simplify to be 1 multiplied by 7 + 10 + 13 + 16, which is going to give us a final sum of 46, and this is going to be the right Riemann sum for the given function F of X. And with that being said, the overall solution to this problem is going to be D. 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.

Left and right Riemann sums Complete the following steps for the given function, interval, and value of n.
f(x) = x + 1 on [0,4]; n = 4
(d) Calculate the left and right Riemann sums.

Key Concepts

Riemann Sums