5-8. Compute the following estimates of ∫(0 to 8) f(x) dx usin...

Question

5-8. Compute the following estimates of ∫(0 to 8) f(x) dx using the graph in the figure.

Graph of a function with labeled axes, showing a fluctuating curve plotted on a grid.

6. T(4)

Accepted Answer

Hello. In this video, we are going to be using the trapezoidal rule with 4 sub-intervals to estimate the definite integral from 2 to 10 of the function F of X. Now, the function F of X is not given to us, but its graph is provided. But in order to approach this problem, let's go ahead and review the trapezoidal rule. Now, the trapezoidal rule states, or is defined as delta X divided by 2, multiplied by the quantity of the initial value of the function, plus 2 multiplied by the sum of the function at all the intermediate midpoints. Plus the final value of the function. He delta X is defined as the difference of our boundaries B minus A, divided by the amount of subintervals we are working with, and the midpoints that we are going to plug into our function are defined as our lower boundary A plus our index I multiplied by delta X. So, in order to approach this problem, the first thing we're going to want to do is determine the width of our sub-intervals. So we'll need to solve for delta X. Delta X in this case, is going to be our bigger boundary minus our smaller boundary. Now, in the given integral, this integral has a boundary of 2 to 10. So, that means that the difference of the boundaries is going to be 10 minus 2. Then we will divide this by the amount of sub-intervals we are working with, and that is going to be 4. So this is going to simplify to be 8 divided by 4, and this will further simplify to give us a delta X value of 2. Now, the next thing we want to do is you want to determine the endpoints for the initial and final value of the function and all the intermediate endpoints. So, we are going to solve for the 4 endpoints we need. So, starting with the initial value X sub zero, that is going to equal to A, which is our lower boundary, 2, plus the value of the current index I, which is 0, multiply it by delta X value, which is 2, and that is going to give us the initial endpoint, which is going to be 2. Then we will repeat this process for X 1, that is going to be 2 multiplied by the current index, which is 1, multiplied by delta X, which is going to give us a value of 4. And repeating this process, going all the way to X 4, is going to give us the end points, X2 equal to 6, X3 equal to 8, and X 4 equal to 10. Now what we want to do is you want to go ahead and expand the trapezoidal rule with these values. So if we were to expand the trapezoidal rule, the trapezoidal rule would be defined. As T of 4 equal to delta X, which is 2 divided by 2, multiplied by F of X of 0, which is F of 2, plus 2 multiplied. By FF 4 Plus F of 6. Plus F of 8. Plus the final value of the function, which will be F of 10. Now, what we want to do is you want to fill in these missing values by using the given graph. Now, based off of the graph, the initial value of the function in F of 2 is going to be 5. Sorry, I apologize. The initial value is going to be 7. Then we will add 2 multiplied by the value of the function at 4, which is going to be 6. Plus 8 + 10. Oh sorry, plus 5 for the value 6. And then plus 9 for the value of 10. And Delta X divided by 2 was simplified to be 1. So, we are left with 7 + 2 multiplied by 6 + 8 + 5. That is going to simplify to be 7 + 2 multiplied by 19 plus 9. And this is going to further simplify to give us a final value of 54. So this is going to be the estimated value of the given functions graph 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 approach this problem, and we will go ahead and see you all in the next video.

5-8. Compute the following estimates of ∫(0 to 8) f(x) dx using the graph in the figure.

6. T(4)

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5-8. Compute the following estimates of ∫(0 to 8) f(x) dx using the graph in the figure.6. T(4)

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5-8. Compute the following estimates of ∫(0 to 8) f(x) dx using the graph in the figure.

6. T(4)