{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.

∫₀² (𝓍²―2) d𝓍 ; n = 4

Accepted Answer

Welcome back, everyone. In this problem, we want to calculate the left and right Riemann sums for the integral of 2 X2 minus 3 with respect to X on the interval from 1 to 3, using N equals 4 subintervals. Now, if we're going to find our Riemann sums, the first thing we need to do is to find the width of our interval, OK? And the width of our interval is equal to the length of the interval. Divided by the number of subintervals and. No, since our interval is from X equals 1 to X equals 3, then its length will be 3 minus 1, and we're dividing that by N equals 4. That's gonna be 2/4 or a half. Know that we have the width of the interval, then we can evaluate the function at each endpoint, OK? So evaluating the function at each endpoint. Mm. And just to be explicit here, we're saying our function F of X equals 2 X2 minus 3, OK. Then we could do that by just writing or creating a table here. So we're gonna have our endpoints and our functions, OK? And we're gonna do it for all of our intervals. Or 4 or end points 4 or 4 subintervals. So now we know we're starting at 1 and each interval has a width of a half. So X0 will be equal to 1, X1 equals 1.5, X2 equals 2, X3 equals 2.5. And X4 will be equal to 3. So now we're going to evaluate FF X equals 2X2 minus 3 at each point. So, for example, when X equals 1, FX equals 2 multiplied by 12 minus 3. Which equals -1. When X1 equals 1.5, then F of X will be equal to 1.5. OK. Again, we're just substituting our value for X into our function. When X equals 2, F X equals 5, when X equals 2.5, F X equals 9.5. And when X equals 3 F X equals 15. OK. So these are our functions, the value of our functions at each of our endpoints. Now that we have that, then we can calculate the left and the right Ryman sums. Because for the left right on some. On the left Riemann sum L4 because of the four subintervals, we're going to we're going to multiply the width of each interval delta X by all of the left end points. So that's F of X0 plus F of X1 plus F of X2 plus F of X3. Not including F of X4 because X4 is not a left end point, OK? So now in this case, that's gonna be equal to 1/2. OK. Multiplied by -1 + 1.5 + 5 + 9.5. And that's a half multiplied by 15, which equals 7.5. So L4, our left Riemann sum is 7.5. Now let's apply the same idea to help us find or right rhyme and so. I know the difference is that all we're going to do is to use the right end points. So let's call that R4. So no, that means we're going to be using F of X1, F of X2, F of X3, and F of X4, but not F of X not because as you guessed it, It is going to be, sorry, X is a, is not a right end point, OK? So no, that's gonna be a half of 1.5 + 5 + 9.5 + 15. Now when we calculate that we get it to be equal to a half of 31 or 15.5. Thus, our left Riemann sum is 7.5 and the right Riemann sum is 15.5. 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.

∫₀² (𝓍²―2) d𝓍 ; n = 4

Key Concepts

Riemann Sums

Definite Integral

Subintervals and n

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