{Use of Tech} Approximating definite integrals

Question

{Use of Tech} Approximating definite integrals Complete the following steps for the given integral and the given value of n.

(c) Calculate the left and right Riemann sums for the given value of n.

∫₁⁷ 1/𝓍 d𝓍 ; n = 6

Accepted Answer

Welcome back, everyone. In this problem, we want to calculate the left and right Riemann sums for the integral of 6 divided by X with respect to X between 2 and 8 using N equals 6 subintervals. Now if we're going to find the left and the right Riemann sums, then it would be first for us to it would be good for us to first find the width of each interval and width is going to be equal to the length of the interval. OK, divided by the number of subintervals in. So in that case, the length of our interval will be 8 minus 2, and we know N equals 66 divided by 6 equals 1. So each interval is 1 unit long. So, for us to find the Riemann sums, it would be good for us to make a note of each endpoint and the function at that endpoint. OK? So let's just create a table for each endpoint and the function at each endpoint where our function is equal to 6 divided by X. Now, in this case here, let me just put our table together. And we know that we have 6 sub-intervals starting at 1 and ending at 8. So X1 X not, sorry. is going to be equal to 2, X1 is 3, X2 is 4, X 35, X 46. X 57, and X 68. OK. And now, all we need to do is to figure out a function at each of these end points. So when X is, sorry, when X is 2, F of X is 6 divided by 2 or 3. Likewise, when it's 3, FF X is 2, when X is 4, FF X is 3 halves, when X is 5, FF X is 6/5. When X is 6, FF X is 1, when it's 7, FF X is 6/7, and when it's 8, FF X is 3 quarters. So no, we have, we've evaluated our function at all of our endpoints and we can use that to help us find our rhyme and sums. Now starting with the left raman sum. OK. The left Ramanson. Is going to be equal to the width of our interval delta X multiplied by all of our left end points. So that is where X equals 2. 3 4, OK. 5 6 And 7, OK, but excluding F of 8 because it is not a left end point. So in this case, we're multiplying 1 by the sum of 3 + 2 + 3 halves plus 6/5 + 1 + 6/7. And now when we go ahead and multiply here, then we should get our left riman some L. To be equal to 669 divided by 70. On the other hand, or or write rhyme and some. It's basically the same thing but with all of our right end points, so we're multiplying our our width the the our interval with one by all of our right end points. So from X1 all the way to X6, but in, but excluding sorry X not OK, because. We know that X0 is not a right end point. So in this case, this is gonna be equal to 1 multiplied by 2 + 3 halves, plus 6/5 + 1 + 6/7 plus 3/4. I know when we multiply that here, then our right Riemann sum is going to be equal to 1,0023 divided by 40. Thus, the left Raman sum is 669 divided by 70, and the right is 1,023 divided by 40. Thanks a lot for watching everyone. I hope this video helped.

{Use of Tech} Approximating definite integrals Complete the following steps for the given integral and the given value of n.
(c) Calculate the left and right Riemann sums for the given value of n.
∫₁⁷ 1/𝓍 d𝓍 ; n = 6

Key Concepts

Riemann Sums

Definite Integral

Subintervals and n

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