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

Question

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

(b) Evaluate each sum using a calculator with n = 20, 50, and 100. Use these values to estimate the value of the integral.

∫₀¹ cos ⁻¹ 𝓍 d𝓍

Accepted Answer

Hello. In this video, we are going to be approximating the value of the definite integral of 0 to 1 of sine inverse of XDX by using the right endpoint Ryman sums with n is equal to 2050, and 100. Then we want to choose which of the values is the best estimate for the integral. OK, in order to approach this problem, we are going to use the definition of the definite integral. The definite integral from A to B of F of XDX is defined as the infinite summation starting from 1 and going to N of F of Xi multiplied by the change in X which is delta X. Here, delta X is defined as B minus A divided by N and Xi is defined as A plus the index I. Multiplied by Delta X. So in order to approach this problem, the first thing we want to do is we want to define the general change in X for all the values of N. Now, the definite integral given to us is from 0 to 1 of sine inverse of X. So A is going to be defined as 0, and B is defined as 1, and that means that our change in X or delta X is going to be B minus A, which is 1 minus 0 divided by N, which is going to simplify to 1 divided by N. This is going to be the general delta X that we use for each of the values of N. Next, what we need to do is we need to rewrite X into terms of X I. Xi is defined as our lower boundary A, which is 0, plus the index I multiplied by delta X, which is going to be 1 divided by N. This will simplify to be I divided by N, and this is going to be the X of I that we plug into the summation. Now what we are going to do is we are going to rewrite the given definite integral into a summation. So we're given the integral, the integral from 0 to 1. Of sine inverse of XDX. First, we are going to rewrite the integral symbol into a summation symbol. That is going to be the summation from I is equal to 1 ton. Then we are going to replace the X with X sub I. That is going to give us sine inverse of I divided by N. And finally, we're going to change DX into Delta X, and that is going to be 1 divided by N. So this is going to be the general summation that we are going to use for each of the values of N, and we want to plug in three values for N. N is equal to 2050, and 100. So, by starting with N is equal to 20, if we were to plug this into the summation, we will have the summation from I is equal to 1 to 20 of sine inverse. Of I divided by 20 multiplied by 1 divided by 20, and by using the help of a calculator, this is going to approximate to be 0.6131. This is going to be the value for N is equal to 20, N is one of the solutions we are looking for. Then we are going to repeat this process with N is equal to 50. By plugging in 50 for N, we will have the summation from Is equal to 1 to 50 of sine inverse of I divided by 50 multiplied by 1 divided by 50. And this is going to approximate to be 0.5873. Which is the second solution we are looking for? Finally, we are going to plug in 100 for N. This is going to give us the summation from Is equal to 1 to 100 of sine inverse of I divided by 100 multiplied by 1 divided by 100, and this will approximate to be 0.5789. So this is going to be the 3 values that we are looking for, for the 3 different values of N, but now we need to answer the last part of this question, which is asking us which value of N is going to be the most accurate for the integral. Well, that value of N is going to be 100. 100 is the most accurate for the value of the integral, since it is the largest interval underneath the integral. Whenever we are working with summations, we always want to choose the largest the largest intervals, since this is going to give us the most accurate result. And with that being said, the solution to this problem is going to be a. 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.

{Use of Tech} Approximating definite integrals with a calculator Consider the following definite integrals.
(b) Evaluate each sum using a calculator with n = 20, 50, and 100. Use these values to estimate the value of the integral.

∫₀¹ cos ⁻¹ 𝓍 d𝓍

Key Concepts

Definite Integrals

Numerical Integration

Limit of Riemann Sums

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