66–71. {Use of Tech} Estimating error Refer to Theorem 8.1 in ...

Question

66–71. {Use of Tech} Estimating error Refer to Theorem 8.1 in the following exercises.

71. Let f(x) = √(sin x).

a. Find a Simpson's Rule approximation to the integral from 1 to 2 of √(sin x) dx using n = 20 subintervals.

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says to approximate the interval from 0 to 1 of the square root of the quantity of cosine of X and quantity DX using Simpson's rule with equal to 20 subintervals. Round your answer to 4 decimal places, and we're given 4 possible choices as our answers. For choice A, we have 0.8733. For choice B, we have 0.8988. For choice C, we have 0.9140. And for choice D we have 0.9263. So, we're asked to approximate our interval here using Simpson's rule with N equal to 20 subintervals. So here we need to recall our Simpson's rule. So, the value of our Simpson rule with equal to 20 sub-intervals, we'll call that S of 20. And recall that this is going to be equal to the quantity of F of X knot. Plus 4 multiplied by F of X1. Plus 2 multiplied by F of X2. Plus 4 multiplied by F of X3. Plus continuous pattern of alternating the coefficients between 4 and 2 until we have plus 4 multiplied by F of X. K minus 1 Plus F of X K. In quantity, and this gets multiplied by delta X divided by 3. So we first need to find our subinterval with delta X. And so recall that delta X is going to be equal to, and here we'll have the upper limit of integration, which we have as 1 minus the lower limit, which is 0, and that quantity is going to be divided by n, which in our case is 20. And so simplify this expression, we see that delta X is going to be equal to 1 divided by 20. Next, we need to find our different grid points, which will denote as X of K. we call to determine the grid points, this is just going to be equal to the lower bound of our limits of integration, which in our case is going to be 0, plus, K multiplied by Delta X, which in our case is 1 divided by 20. And finally, we need to identify F of X of K. And this is just going to be equal to the integrand of our integration. So we'll have the square root. Of the quantity of cosine of X K in quantity or F of X of K. So now we need to calculate our values of XK and F of XK to use in our Simpson rule. And this is best done using a table. And so on the screen, we have a table. That has 3 columns. The first column has K and it has the values going from 0 to 20. The next column is xk, and using our formula for xk, we get the values of 0, 0.05, 0.10, 0.15, 0.20, 0.25, 0.30, 0.35, 0.40, 0. 45 0.50, 0.55 0.60, 0.65, 0.70, 0.75, 0.80, 0.85, 0.90, 0.95, and 1. The last column is labeled F of XK. And so here we're going to be evaluating our function F of XK for each value of XK. In doing so, we'll get the values of 1 0.9993749, 0.9974990, 0.9943697 0.98. 99831. 0.9843335 0.9774132 0.9692124 0.9597192 0.9489189. 0.9367938. 0.9233225, 0.9084798, 0.8922353. 0.8745526, 0.8553881, 0.8346896. 0.8123935 0.7884225 0.7626815 and finally 0.7350526. So we'll use these values to calculate our simson role. In doing so, we'll see that S of 20 is going to be equal to the quantity of 1 + 4 multiplied by 0.999. 374 9. Plus 2 multiplied by 0.9974990. Plus, and we'll continue using our formula until we reach the very end of our sequence. So that last two. Terms will be + 4 multiplied by the quantity, and here we'll have to look at. X 19, and the value of F of X. 19 is going to be 0.762. 6815 and then we'll have plus for the last term 0.735 0526 in quantity and this gets multiplied by delta X divided by 3. And so delta X is going to be the quantity of 1 divided by 20 in quantity and that gets divided by 3. And so evaluating. This expression using our calculator, we see that S of 20 is going to be equal to. 0.9140 and here we have rounded to 4 decimal places. So this is the answer that we're looking to find for the approximation of our integral using the Simpson rule. And if we compare this answer to our answer choices, we see that the correct answer choice corresponds to answer choice C. So I hope that this explanation has been useful, and I'll see you in the next video.

66–71. {Use of Tech} Estimating error Refer to Theorem 8.1 in the following exercises.71. Let f(x) = √(sin x).a. Find a Simpson's Rule approximation to the integral from 1 to 2 of √(sin x) dx using n = 20 subintervals.

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66–71. {Use of Tech} Estimating error Refer to Theorem 8.1 in the following exercises.
71. Let f(x) = √(sin x).
a. Find a Simpson's Rule approximation to the integral from 1 to 2 of √(sin x) dx using n = 20 subintervals.