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.

66. Let f(x) = cos(x²).

a. Find a Midpoint Rule approximation to ∫[-1 to 1] cos(x²) dx using n = 30 subintervals.

Accepted Answer

Hello. In this video, we are going to be using the midpoint rule approximation with 20 sub-intervals to approximate the integral from 2 to 4 of sin of X 2 DX. We want to round our answer to 4 decimal places. In order to approach this problem, let's just quickly review the definition of the midpoint rule. The midpoint rule is defined as the sum from I is equal to 1 ton of F of Xi delta X, and this will expand to delta X multiplied by F of X1 plus F of X2 all the way. To F of X N. Now, in this case here, Delta X is defined as the difference of our boundaries divided by the sub-intervals, and our endpoints Xi for the midpoint rule is defined as our lower boundary A plus the quantity I minus 1/2 multiplied by delta X. So, in order to approach this problem, there are a few things we need to determine. First, let's go ahead and determine our delta X value. Now, in the problem, we are working with the integral on the boundary from 2 to 4. So, the difference of the boundaries for delta X is going to be the upper minus lower, which is 4 minus 2. Now, the problem tells us that we are working with 20 sub intervals, so this is going to be divided by 20. This is going to simplify to be 1 divided by 10, which is going to be the value of delta X that we are going to work with for the midpoint rule. Now, for the midpoint rule, we first need to determine our midpoint, our midpoints to plug into the function. Now, in the given integral, the function is given to us as F of X is equal to sine of X squared. So, for the first midpoint, X1. That is going to equal to our lower boundary, which is 2, plus the quantity, I minus 1/2, which is the value of the current index, so that is going to be 1 minus 1/2 multiplied by the value of delta X, which is 1/10. This is going to simplify to give us the value of 2.05. Then what we would need to do with this midpoint is we would need to plug it into our function afterward. So F of 2.05 is equal to sine of 2.05 squared, and that is going to give us the value of negative 0.8727987. So this is what we'll need to do for each of the endpoints. But what's a little tedious about this problem is that because we are working with 20 sub-intervals, we would have to repeat this process of finding a midpoint and plugging it into the function 19 more times. So by using a spreadsheet, we can go ahead and organize that data along the side. So what I provided here is the values of all the midpoints or the 20 midpoints we are looking with, and the last column represents the value of that midpoint plugged into our function. Now, this value here is going to be the value of the sum of all 20 midpoint values. So, recall that the midpoint rule is defined as M20 is equal to delta X multiplied. By the sum from 1 to 20 of our function F of Xi. Now, based off the chart here, this value is going to be -0.5548482. So, if we plug in this value of the sum and delta X, we'll be left with 10 multiplied by negative 0.5548482. And by using the help of a calculator, this is going to evaluate to be -0.0555, estimated to 4 decimal places. This is going to be the value of the given definite integral by the midpoint rule. And with that being said, the overall solution to this problem is going to be a. So, I hope this video helps you in understanding how to approach this problem, and we will go ahead and see you all in the next video.

66–71. {Use of Tech} Estimating error Refer to Theorem 8.1 in the following exercises.66. Let f(x) = cos(x²).a. Find a Midpoint Rule approximation to ∫[-1 to 1] cos(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.
66. Let f(x) = cos(x²).
a. Find a Midpoint Rule approximation to ∫[-1 to 1] cos(x²) dx using n = 30 subintervals.