95–98. {Use of Tech} Numerical integration Estimate the follow...

Question

95–98. {Use of Tech} Numerical integration Estimate the following integrals using the Midpoint Rule M(n), the Trapezoidal Rule T(n), and Simpson’s Rule S(n) for the given values of n.

97. ∫ (from 0 to 1) tan(x²) dx; n = 40

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says to estimate the interval from 0 to 2 of sine of X cubed DX. Using Simpson's rules of N with N equal to 20, round your answer to two decimal places, and we're given four possible choices as our answers. For choice A, we have 0.38. Choice B, we have 0.45. Your choice C was 0.52, and for choice D, we have 0.61. So we're asked to estimate or an integral here using Simpson's rule. So recall for a Simpson's rule that we have S of N, and that 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 X. 3. Plus, and this competit continues of alternating the coefficient between 4 and 2, until we get to the end, and we will have plus 4 multiplied by F of X N minus 1. Plus F of XN. In quantity And so here in this formula, N has to be an even integer. Which is what we have in this case since N is equal to 20. So now we just need to identify a couple of other quantities. The first quantity that we need to identify is going to be delta X, which is going to be our subinterval width. Recall that delta X is going to be equal to our upper limit of integration, which is 2 minus our lower limit of integration of 0, and then that quantity is going to be divided by N, which in our case is 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 identify our grid points, which will denote as X K. And here we call that for X of K, which is our good points, that that's going to be equal to our lower limit of integration, which is going to be 0, plus K multiplied by delta X, which is equal to 1 divided by 10. And so when we simplify this expression, we see that X of K is just going to be equal to K divided by 10. And finally, we just need to identify our function F of X, and this is just going to be equal to the integram of our integral, which is going to be equal to the sine of X cubed. So now what we need to do is to find each of our grid points, X sub K, and then evaluate F of X at those grid points. And so this information is best kept in a table. So, on the screen here, we have created a table that has 3 columns. The first column is denoted as K, and in this column, we have the integers starting from 0 and going to 20, that N is equal to 20. In the next column, we will have XK, and here we will use our formula for XK to find the value of XK for each value of K. In doing so, in this column, we will have the values of 0, 0.1, 0.2, 0.3. 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1, 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'll have F of XK. So we'll need to evaluate F of X for each value of XK in the previous column. In doing so, in this column, we'll get the values of 0, 0.001,0000. 0.00799999. 0.0269967. 0.0639563. 0.1246747. 0.2143 243. 0.3363139. 0.4899218, 0.6661241. 0.8414710. 0.9713864. 0.9876689. 0.8102583. 0.3871999. -0.2312938. Minus 0.8159713 minus 0.9799450 minus 0.4360325. 0.5445183 and 0.9893582. So now that we have calculated all these values, we can actually apply them and calculate our estimate of this interval using 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 delta X, which is 1 divided by 10, and that quantity is going to be divided by 3. Then that gets multiplied by the quantity of F of X, which we just calculated to be zero. You'll have plus 4 multiplied by F of X1, which is 0.001,0000. Then we'll have plus 2 multiplied by F of X2, which is 0.0079999. Then we'll have + 4 multiplied by F of X3, which is 0.026. 99 67. Then we'll have plus, and will continuous pattern of alternating the coefficient between 4 and 2 multiplied by F of X of K. Until we get to the end of the pattern, and then we'll have loss. 4 multiplied by F of X 19, which is 0.5445183, then we'll just have plus F of X20, which is 0.9893582 in quantity. And so, evaluating this expression, using our calculator, we'll see that S20 is going to be equal to. 0.45. So here we just round to two decimal places, and this is the answer that we're looking for for this problem. And if we compare our answer now to our answer choices, we see that the correct answer choice corresponds to answer choice B. So I hope that this explanation has been useful, and I'll see you in the next video.

95–98. {Use of Tech} Numerical integration Estimate the following integrals using the Midpoint Rule M(n), the Trapezoidal Rule T(n), and Simpson’s Rule S(n) for the given values of n.97. ∫ (from 0 to 1) tan(x²) dx; n = 40

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95–98. {Use of Tech} Numerical integration Estimate the following integrals using the Midpoint Rule M(n), the Trapezoidal Rule T(n), and Simpson’s Rule S(n) for the given values of n.
97. ∫ (from 0 to 1) tan(x²) dx; n = 40