{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.

∫₃⁶ (1―2𝓍) d𝓍 ; n = 6

Accepted Answer

Welcome back, everyone. In this problem, we want to calculate the left and right Riemann sums for the definite integral of 2 minus X with respect to X between the limits of 1 and 4 with N equals 6 subintervals. Now, if we're going to find the Riemann sums, let's first start by finding the width of each interval, OK? Now, the width of each interval X, OK, or Delta X is going to be equal to the length of the interval. Divided by the number of subintervals in. Now in this case, our interval goes from 1 to 4, so its length is 4 minus 1, and we know there are 6 subintervals. Thus, the width of an interval is 36 or 2. Know that we have the length of an interval, then we can list the sub-interval endpoints and evaluate the function at each endpoint. So, I'm gonna create a table here, OK, of the endpoint and the function at the endpoint, OK? And we know our function is 2 minus X. So now if we just create a table of that here. OK. And we make a note of all of all our endpoints. So, X, X1, uh, let me write it over here. X X1 X2. X3, 4, X5, and X6, then we can evaluate the function and each. Now X0 will be equal to 1, where our limit starts or where interval starts, which means F of X will be 2 minus 1 or 1. X1 is 1.5, which means F of X at that point will be 0.5, X2 is 2, which means it's gonna be 0, because remember, the width of the interval is a half, so that's why we're going up by a half each time. X3 is 2.5 and FFX would be negative 0.5, X4 is 3, then FF X would be -1. X5 is 3.5, so FF X is -1.5, and X6 is 4, so FF X is -2. So now we have each of our endpoints and the value of the function at each of the endpoints. So now that we have this information, then we can compute the Riemann sum. Now starting with the left, Ray and some. Left to him and some. Remember that it's going to be the sum of all our endpoints or left endpoints rather, multiplied by the width of the interval. So that's really delta X multiplied by starting from the left F of X0 plus F of X1 plus F of X2 plus F of X3, OK, plus F of X4. Plus F of X5, OK, but not including F of X6 because remember X6 is not a left end point. So now, when we substitute all of our values that we know, then that would be a half of 0, sorry, a half of 1, that was the first one, plus 0.5 plus 0, minus 0.5 minus 1, minus 1.5, OK? In this case here we can see that 1 minus 1 is 0, 0.5 minus 0.5 is 0, so we're left with a half of -1.5, which would be equal to -3/4. So that is the left Riemann sum L6. Next, let's use the same idea to help us find the right rhyme and song. Now, to find the right Riemann sum, all we're going to do is to use the right end of points, OK? So, essentially, here, we're using all of these points, OK, from X1 to X6, but not X not because that is a left end point. That's only a left end point. So now, in this case, you applying the same idea, that's the width intervalla half multiplied by 0.5 plus 0, minus 0.5 minus 1. -1.5 minus 2, OK. And now, in this case, this is gonna turn out to be a half of -4.5, which is gonna be equal to -9/4. So our left Riemann sum is -3/4 and our left or right Riemann sum are 6. Is going to be -9/4. 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.

∫₃⁶ (1―2𝓍) d𝓍 ; n = 6

Key Concepts

Definite Integral

Riemann Sum

Left and Right Riemann Sums

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