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.

70. Let f(x) = e^(-x²).

a. Find a Simpson's Rule approximation to the integral from 0 to 3 of e^(-x²) dx using n = 30 subintervals.

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says to approximate the interval from 0 to 2 of E rates to the quantity of minus X cubed in quantity DX. Using Simpson's rule with equal to 20 subintervals, run your answer to 4 decimal places, and we're given 4 possible choices as our answers. For choice A, we have 0.8930. For choice B, we have 0.9283. For choice C, we have 1.0288. And for choice D, we have 1.0463. So we're asked to approximate our integral using our Simpson rule. So, we call for the Simpson rule that we have S of N. And this is going to be equal to. Delta X divided by 3, multiplied by the quantity of F of X0. Plus 4 multiplied by F of X1. Plus 2 multiplied by F of X2. Plus 4 multiplied by F of X3. Plus, and this pattern continues of alternating between the coefficient of 4 and 2, until we get to plus. 4 multiplied by F of X N minus 1. Then we'll have less F of XN. In quantity So to apply the Simpson rule, we first need to determine our sub-interval width, which is delta X. Recall that delta X is going to be equal to the quantity of the upper limit of integration, which in our case is 2, minus the lower limit, which is 0, and then that quantity is divided by n, which in our case we're told that n is going to be equal to 20. And so when we evaluate this expression, we see that Delta X is going to be equal to 1 divided by 10. Next, we need to determine our grid points, which will denote as X K. And here we call your formula for finding the grid points. We call that's going to be equal to our lower limit of integration, which in our case is 0, plus K multiplied by delta X, which in our case is 1 divided by 10. And so when we simplify this expression, we see that X of K is going to be equal to K divided by 10. And finally, we just need to identify our function F of X. And F of X is just going to be the integrand of our integrals. So this is going to be equal to E raised to the quantity of minus X cubed in quantity. So now what we want to do is to find all of our grid points and then evaluate F of Xs at each of those grid points. And this information is best kept in a table. So, on the screen here, we will create a table that has 3 columns. The first column is K. And K here has the integer values going from 0 to 20 in that column. The second column is xk, and so here we'll use our formula to calculate, to calculate XK for each value of K. In doing so, in this column, we'll get the values of 0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.91. 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9, and 2. And in the final column we have F of XK. And so here we're just going to evaluate our function F of X for each value of XK that we just found. In doing so in this column, we will get the values of 1 0.9990005. 0.9920319. 0.9733612. 0.9380050. 0.8824969. 0.80573553. 0.7096382. 0.5992958. 0.4823911. 0.367 8794. 0.2642129. 0.1776393 0.1111361 0.0643126. 0.0342181. 0.0166391. 0.0073504 0.0029322 0.0010500 and 0.0003355. So now we have those values, we can actually calculate our Simpson rule. So here we have N equal to 20, so we're looking for S of 20. And so this is going to be equal to our delta X value, which is going to be 1 divided by 10, then that quantity is going to be divided by 3, and then that gets multiplied by the quantity, and here we have F of X0, which is 1, then we'll have plus 4 multiplied by F of X1, which has a value of 0.999. 0005, then we'll have plus 2 multiplied by F of X2, which is 0.992. 0319, it will have plus 4 multiplied by F of X3, which is 0.9733. 612. Then we'll have plus, and again, continuing the pattern of alternating the coefficient between 4 and 2. Multiplied by the F value. At the X K. Location, continuing on that pattern until we reach the end, and we'll have less 4 multiplied by F of X19, which is going to be 0.001,0500. Then we'll have plus F of X20, which is 0.000. 3355. In quantity And so, we can evaluate this expression using our calculator, and we see that S20 is going to be equal to. 0.8930. So this here is the answer that we're looking for for this problem. And if we compare our answer here to our answer choices, we see that the correct answer choice corresponds to answer choice A. 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.70. Let f(x) = e^(-x²).a. Find a Simpson's Rule approximation to the integral from 0 to 3 of e^(-x²) dx using n = 30 subintervals.

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66–71. {Use of Tech} Estimating error Refer to Theorem 8.1 in the following exercises.
70. Let f(x) = e^(-x²).
a. Find a Simpson's Rule approximation to the integral from 0 to 3 of e^(-x²) dx using n = 30 subintervals.