23-26. {Use of Tech} Simpson's Rule approximations. Find the i...

Question

23-26. {Use of Tech} Simpson's Rule approximations. Find the indicated Simpson's Rule approximations to the following integrals.

24. ∫(4 to 8) √x dx using n = 4 and n = 8 subintervals

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says to estimate the interval from 2 to 6 of the square root of the quantity of X2 + 1 in quantity DX using symptoms rule for N equal to 4 subintervals. Round your answer to 2 decimal places. And we're given 4 possible choices as our answers. For choice A, we have 16.54. For choice B, we have 14.65. For choice C, we have 12.64, and for choice C, we have 15.64. So we're asked to estimate our integral here using Simpson's rule with an equal to 4 sub-intervals. So, we call for the Simpson's rule that the interval from A to B, of F of X. DX is approximately equal to Delta X divided by 3, multiplied by the quantity, and for N equal to 4, we're going to have F of X0. Plus 4 multiplied by F of X1. Plus, 2 multiplied by F of X2. Plus Or multiplied by F of X3. Plus F of X or in quantity. So, our first step is to find our self-interval width, which is delta X, and we call the delta X, it's just going to be equal to the quantity of B minus A in quantity divided by N. Which in our case is going to be equal to. The quantity of 6 minus 2. In quantity divided by 4. Which is equal to one when we evaluate that expression. So now that we have Delta X, we can determine our values for X of I. And so X knot is just gonna be equal to the lower bound of our limits of integration, which is going to be equal to 2. And then we'll just need to add delta X to each value to get the next value in our sequence for X of I. So that means that X of 1 is going to be equal to 3, X2 is going to be equal to 4. X3 is going to be equal to 5. And X4 is going to be equal to 6. And we're now going to need to evaluate our function F of X at each one of these X values. So, for our first one, we'll have F of X0 and F of X here is just our integram, which is the square root of the quantity of X2 + 1. So, this is going to be equal to F of 2, which is equal to the square root. Of the quantity of 2 squared plus 1 in quantity which is equal to the square root of 5. Next, we need to evaluate F of X1, which is equal to F of 3. Which is equal to the square root. Of the quantity of 3 squared plus 1. In quantity, which is equal to the square root of 10. Next, we'll look at F of X2. And this is going to be equal to F of 4. Which is equal to the square root of the quantity of 4 quad plus 1 in quantity, which is going to be equal to the square root of 17. Next, we'll look at F of X3. Which is going to be equal to F of 5. Which is equal to the square root of the quantity of 52 plus 1. Which is equal to the square root of 26. And finally, we'll look at F of X4. Which is equal to F of 6. Which is equal to the square root of the quantity of 6 squared plus 1 in quantity, which is equal to the square root of 37. So now we can apply our Simpson rule to our integrals. So, the integral. From 2 to 6 of the square root of the quantity of X2 plus 1 in quantity DX is going to be approximately equal to, and here for delta X, recall that we have 1, so we'll have 1 divided by 3, multiplied by the quantity, and here we'll have the square root of 5, plus 4 multiplied by the square root of 10. Plus 2 multiplied by the square root of 17. Plus, or multiplied by the square root of 26. Plus, the square root of 37. In quantity. And when we evaluate this expression and round it to two decimal places, we see that this is going to be approximately equal to. 16.54. So this here is the answer that we're looking for for this problem. And if we compare that 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.

23-26. {Use of Tech} Simpson's Rule approximations. Find the indicated Simpson's Rule approximations to the following integrals.
24. ∫(4 to 8) √x dx using n = 4 and n = 8 subintervals

Key Concepts

Simpson's Rule

Definite Integral

Subintervals

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