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.

69. Let f(x) = sin(eˣ).

a. Find a Trapezoid Rule approximation to ∫[0 to 1] sin(eˣ) dx using n = 40 subintervals.

Accepted Answer

Hello. In this video, we are going to be using the trapezoid rule approximation with 20 sub-intervals to approximate the definite integral from 1 to 2 of cosine to the 2 to the power of X. We want to round our answer to 4 decimal places. So just by looking at the given given integral, we are working on the bounds from 1 to 2. The function that we are going to be working with is cosine of 2 of X. And we are asked to work with 20 subintervals. Now the trapezoidal rule is defined as the following. The trapezoid rule is Delta X divided by 2 multiplied by the initial value of our function F of X subzero, plus 2 multiplied by the sum from I is equal to 1 ton of F of XI. Plus, the final value of our function F of X sub N. Here, the end points we plug into the function X sub I are defined as our lower boundary A plus I multiplied by delta X, and delta X is defined as the difference of our boundaries divided by the amount of our sub-intervals we're working with. So, in order to approach this problem, the first thing we're going to do is we're going to determine our delta X value. Delta X is going to be the difference of the boundaries, which is bigger minus smaller, so that will be 2 minus 1, and we will divide this value by the amount of sub-intervals we're working with, which is going to be 20. This will simplify to be one divided by 20, which is going to be the value of delta X we will work with. Now, in order to determine the initial value of our function, the first thing we need to do is we need to determine the initial value or the initial midpoint. So, in order to find this value, we're going to find X sub zero. That will be defined as our lower boundary, which is 1, plus the index value, which is 0, multiplied by delta X, which is 1 divided by 20, and this is going to give us a value of 1. Then, in order to plug this into the trapezoidal rule, we will have to plug the value of one into the function. So that is going to be F1 equal to cosine of two rates to the power of one, and that is going to give us a decimal value of -0.4161468. Then we will need to repeat this process for all the end points in between the initial and final value, and the final value itself. But this is going to be very tedious because we were going to have to repeat this process. Of finding an endpoint and plugging it into a function 19 more times. So, let's go ahead and organize that data into a table. So what I provided here is a table that gets all the values that we need for the trapezoid rule. The middle column represents the values of all the end points that we are going to be plugging into the trapezoidal rule. This very first point at one is the initial value, the very last point is going to be the final value of the function, and all the middle values are all the middle end points between the initial and final midpoint. Now, the third column here represents the value of the function at each of these end points, and notice that the sum of all the middle endpoints is -15.7236269. So by using this table, we can go ahead and define the trapezoidal rule. T of 20 for the given definite integral is going to equal to delta X divided by 2, which is going to be 1 divided by 20 divided by 2 multiplied by the initial value of the function, which is going to be this first value. That is going to be negative 0.4161468. Then we will add 2 multiplied by the sum of all the media values, which is going to be -15.7236269. Then we will add the final value of the function, which is given to us in the very last bubble here, and that is going to be negative 0.6536436. Now, by multiplying and using a calculator, this is going to give us a final value of negative 0.8129, rounded to four decimal places, and this is going to be the value of the definite integral by using the trapezoidal rule. And with that being said, the overall solution to this problem is going to be B. So, I hope this video helps you in understanding how to use 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.69. Let f(x) = sin(eˣ).a. Find a Trapezoid Rule approximation to ∫[0 to 1] sin(eˣ) dx using n = 40 subintervals.

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66–71. {Use of Tech} Estimating error Refer to Theorem 8.1 in the following exercises.
69. Let f(x) = sin(eˣ).
a. Find a Trapezoid Rule approximation to ∫[0 to 1] sin(eˣ) dx using n = 40 subintervals.