{Use of Tech} Approximating definite integrals with a ...

Question

{Use of Tech} Approximating definite integrals with a calculator Consider the following definite integrals.

(a) Write the left and right Riemann sums in sigma notation for an arbitrary value of n.

∫₀¹ cos ⁻¹ 𝓍 d𝓍

Accepted Answer

Welcome back, everyone. In this problem, we want to write the left and right Riemann sums in sigma notation for an arbitrary value of N to approximate the definite integral of the square root of 1 + X squared with respect to X between 2 and 4. Now, let's first make note of the information that we have here. So, for starters, we know that the interval, OK, is from 2 to 4, OK? Which means that A equals 2. And B equals 4, OK. We also know That FFX, our problem here is gonna be the function that is being integrated, so that is the square root of 1 plus X squared. Now what do we know about the left and the right Riemann sums in sigma notation? Well, recall. That in sigma notation. The Riemann sum is going to be the sum from I equal in this I equals 1 all the way up to N of F of Xi multiplied by. Delta X, OK, where Xi represents the end points and delta X represents the width of each subinterval. If we can find the width of each subinterval which will remain the same, then we can use that in our left Riemann sum with our left end points and then in our right Riemann sum with the right end points. Now let's first find the width of each subinterval. We know that that's going to be equal to B minus A divided by N. That's 4 minus 2 divided by N or 2 divided by n. Now that we have that we can find ourselves. Starting with the left rhyme and some, OK, so far. The left Ryman song. OK. Then the left end points. The left End points XI Is going to be equal to 2 + 1 minus 1 multiplied by the width of each subinterval. So in this case, that's going to be equal to 2 plus I minus 1 multiplied by 2 divided by N. And that is going to be equal to 2 + 2 multiplied by i minus 1 divided by n. And now that we have the left end points, that means the left Riemann sum is going to be the sum from I equals 1 ton of the square root. So now we're going to put in F of X, and anywhere we see X, we're going to replace it with the left end points. So that's 1 plus. X 2 R 2 + 2 multiplied by i minus 1 divided by N squared. And that will be multiplied by the width of our sub interval 2 divided by n. Thus this is the left Riemann sum. Next, let's move on to the right rhyme answer so, OK. And for the right rhymansum. OK. Then the right end points. OK. Are going to be. Equal to 2 + i multiplied by the width of each subinterval. So that's gonna be 2 + i divided by N multiplied by 2 or 2 + 2 I divided by N. I know that we have that, that means the right Riemann sum is going to be equal to the sum from I ton of the square root of 1 + X or 2 + 2 I divided by n squared multiplied by 2 divided by n. Thus, the right Riemann sum is here written in red and the left, sorry, the the left Riemann sum is written in red, OK. And the right rhyme and sum is here written in blue. Thanks a lot for watching everyone. I hope this video helped.

{Use of Tech} Approximating definite integrals with a calculator Consider the following definite integrals.
(a) Write the left and right Riemann sums in sigma notation for an arbitrary value of n.

∫₀¹ cos ⁻¹ 𝓍 d𝓍

Key Concepts

Definite Integrals

Riemann Sums

Sigma Notation

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