In Exercises 11–22, estimate the minimum number of subinterval...

Question

In Exercises 11–22, estimate the minimum number of subintervals needed to approximate the integrals with an error of magnitude less than 10^-4 by (b) Simpson’s Rule. (The integrals in Exercises 11–18 are the integrals from Exercises 1–8.)
∫ from 2 to 4 of 1/(s - 1)² ds

∫ from 2 to 4 of 1/(s - 1)² ds

Accepted Answer

Hello there. Today we are going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Determine the minimum number of subintervals needed so that Simpson's rule approximates the integral, the integral evaluated from 1 to 5 of 1 divided by parentheses of x plus 1 to 2 d x with an error of magnitude less than 10 to -4. OK. So it appears for this particular problem we're ultimately asked to determine what is the minimum number of sub-intervals that are required, such that Simpson's rule will approximate the integral that is provided to us by the prom itself, that has an error of magnitude that is less than 10-4. OK. So now that we know what we're ultimately trying to solve for, let's take a moment to read off our multiple choice answers to see what our final answer may be. A is 18, B is 19, C is 20, and D is 142. OK, so our first step that we need to take is we need to recall and use the error-bound formula for Simpson's rule, which states that the absolute value of capital E subscript S, which I'll call ES for short. So once again the error bound formula for Simpson's rule states that the absolute value of EFS. Is less than or equal to parentheses b minus a to the power of 5, all divided by 180 multiplied by n to the power of 4 multiplied by max as a is less than or equal to x as x is less than or equal to b of f to the power of 4. Of X. OK. So, we need to note that where in this particular case, A will be equal to 1, and B will be equal to 5. And lowercase n will be the, or rather it will represent the number of sub-intervals, which we need to note, this is very important, as we should recall, since we're dealing with Simpson's rule, we need to know. Note that the number of sub-intervals when we are using Simpson's rule will have to be an even value. So N once again is the number of sub-intervals, which must be an even value for this particular problem. And we need to recall and note that F to the power of 4 X is the 4th derivative of the integrand. F of X is equal to 1 divided by parentheses X plus 1 of 2. So with that said, we now need to find the 4th derivative of this function of F of X. So let us note that we can also write our expression for FX. The function FX is equal to x + 1 to the power of -2. So, our first derivative F X will be equal to. 2 multiplied by parentheses of x + 1 to the power of 3. And F of X, the second derivative of FX, will be equal to 6 multiplied by X + 14. And F. Of x, so the third derivative of fx will be equal to -24 multiplied by parentheses of x plus 1 to the power of -5. And finally, the 4th derivative of fxF to the power of 4 x will be equal to 120 multiplied by parentheses of x plus 1 to the power of 6. So, our next step now. So this will be step 2, is we need to determine or rather find the maximum of the absolute value of F 4 X. On the closed interval, which would be square brackets of 1.5, is once again square brackets represent a closed interval. So, with that said, we need to recall and note based on our prior knowledge that we could go ahead and write that F to the power of 4. Of X will be equal to 120 multiplied by parentheses of X + 1 to the power of 6, and we need to note that this function decreases as X increases. So that will mean, let's make a note, this will mean that the maximum will be at X is equal to 1. Fantastic. So, with that said, we could go ahead and write that the absolute value of F 4 X, which in this case we need to note that X is equal to 1, so we need to take the absolute value of F 41 is equal to 120 multiplied by parentheses of 1 + 1 to -6. Which will be equal to 120 multiplied by 2 to the power of -6. Which will be equal to when we write in its most simplified fractional form in the in a fraction, it will be equal to 15 divided by 8. Fantastic. And once again, that's when you write your answer in its most simplified form, meaning you can't simplify it any further. So, moving right along here, our next step now, so this will be step number 3, is we need to plug our values back into our error-bound formula and solve for our value of n. So essentially we need to take the absolute value of eS is less than or equal to 5 minus 1 parentheses to the power of 5, all divided by 180 multiplied by n to the power of 4 multiplied by 15 divided by 8. Which will be less than 10 to -4. So, what we need to do is we are ultimately trying to isolate and solve for n. So when we begin to do that, we will find that we could go ahead and write 45 divided by 180 multiplied by n to the 4 multiplied by 15 divided by 8. is less than 10 to -4. And when we get N all by itself on one side of the expression, we will find that we can write 4 multiplied, or so it's gonna be 4 of 5. So 45 divided by 180, multiplied by 15 divided by 8 multiplied by 10 to -4, which is less than n to the 4. Fantastic. So what we need to do now is we need to take the. Square, so we're gonna have to use a square root, but we need to use a root 4 in order to get rid of this power of 4. So, we will be able to write that N is greater than the 4th root of 4 to 5, so we need to take the 4th root. Of, so once again, the fourth root of 4 to the power of 5 divided by 180, multiplied by 15, divided by. 8 multiplied by, so it's gonna be 8 multiplied by 10 to the power of -4. Fantastic. And once again, we're taking the 4th root in order to get rid of the power of 4 on our value of N. So with that said. We now need to determine what this value of this fourth root is, and we need to note that the fourth root of 45 divided by 180 multiplied by 15 divided by 8 multiplied by 10-4 will be equal to 18.07. But we need to note that since Simpson's rule requires n to be an even number, we need to note that the smallest even integer sufficient to keep Simpson's rule error within the desired limit will have to be in this particular case will be 20 because we need to round to the nearest whole number. And we need to note that it has to be the smallest even integer that is sufficient to keep Simpson's rule error within the desired limit. So, in order to make sure we're within our margin, or I should say magnitude, so once again, trying to determine once again, an error of magnitude that is less than 10-4, that is why our final answer for the sub-intervals has to be 20. So looking back at our multiple choice answers, our correct and final answer without a doubt will be multiple choice answer letter C, which once again is 20. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

In Exercises 11–22, estimate the minimum number of subintervals needed to approximate the integrals with an error of magnitude less than 10^-4 by (b) Simpson’s Rule. (The integrals in Exercises 11–18 are the integrals from Exercises 1–8.)∫ from 2 to 4 of 1/(s - 1)² ds

Watch next

In Exercises 11–22, estimate the minimum number of subintervals needed to approximate the integrals with an error of magnitude less than 10^-4 by (b) Simpson’s Rule. (The integrals in Exercises 11–18 are the integrals from Exercises 1–8.)
∫ from 2 to 4 of 1/(s - 1)² ds