Midpoint Riemann sums Complete the following steps for the giv...

Question

Midpoint Riemann sums Complete the following steps for the given function, interval, and value of n.

ƒ(𝓍) = 1/x on [1,6] ; n = 5

(d) Calculate the midpoint Riemann sum.

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says to calculate the midpoint remand sum for F of X, which is equal to 2 divided by the quantity of X plus 1 in quantity on the close interval from 2 to 7 with N equal to 5 sub-intervals. So the first thing we need to do is determine our sub-interval width. We'll call that sub-interval with delta X. And we call that delta X is going to be equal to, and here we'll have the upper bound of our interval, which is going to be 7, minus our lower bound, which in our case is 2, and that quantity is going to be divided by the number of sub-intervals N, which in our case is 5. And when we evaluate this expression, we see that delta X is going to be equal to 1. Next we need to calculate the midpoint for each of our sub-intervals. So, for our first subinterval, it's midpoint, which we'll call X1, is going to be equal to, and here we call how to calculate the midpoint for the sub-interval. That's gonna be the lower bound of our interval, which is going to be 2, plus, 1 divided by 2 multiplied by delta X. Now, our case delta X is 1, so that means that X1 is going to be equal to 2, plus 1/2 multiplied by 1. Which is going to be equal to. 5 divided by 2. Now, to find the midpoint for the next sub interval, which in our case, we'll call X2. That that is just gonna be equal to X1 plus delta X. And so, X2 is going to be equal to X1, which is going to be 5 divided by 2, plus delta X, which is just 1. And evaluating this expression, we see that X2 is going to be equal to 7 divided by 2. And to find the subsequent midpoints for the following remaining infos, we're just going to add delta X to the previous midpoint. That means that X3 is going to be equal to X2, which is 7 divided by 2, plus delta X, which is 1, and that means that X3 is going to be equal to 9 divided by 2. 4 X4. That's gonna be equal to X3, which is 9 divided by 2, plus delta X which is 1, and this is going to be equal to 11 divided by 2. And for a final midpoint, X5. That's going to be equal to X4, which is 11, divided by 2. Plus Delta X, which is 1, which is going to be equal to 13 divided by 2. So now that we have calculated the midpoints, we can now look at calculating the midpoint remand sum. So, we call for the remand sum that we're going to have the sum of K equal to 1 to 5, since we have 5 subintervals of F of XK. Multiplied by Delta X. Which is going to be equal to the quantity of F of X1. Plus F of X2. Plus F of X3. Plus F of X4. Plus F of X5. In quantity Multiplied by Delta X. So what we need to do is to evaluate our function F of X at each of our midpoints. So, starting with our first midpoint, we'll have F of X1. Which is going to be equal to F of 5 divided by 2. Which is equal to 2 divided by the quantity of 5 divided by 2 plus 1 in quantity. And simplifying this expression, we say that this is going to be equal to 4 divided by 7. Next, we'll calculate F of X2, which is going to be equal to F of 7 divided by 2, which is equal to 2 divided by the quantity of 7 divided by 2 plus 1 in quantity. And when we evaluate this expression, we see that this is going to be equal to 4 divided by 9. The next value that we need to calculate is gonna be F of X3. This is going to be equal to F of 9 divided by 2. Which is going to be equal to 2 divided by the quantity of 9 divided by 2, plus 1 in quantity, and when we evaluate this expression, this is going to be equal to 4 divided by 11. Next, we need to calculate F of X4. Which is going to be equal to F of 11, divided by 2. Which is going to be equal to 2, divided by the quantity of 11 divided by 2. Plus 1 in quantity, and evaluate this expression, this is going to be equal to 4 divided by 13. And finally, we'll need to calculate F of X5. Which is going to be equal to F of 13, divided by 2. Which is going to be equal to 2, divided by the quantity of 13 divided by 2, plus 1 in quantity, and evaluating this expression, this is going to be equal to or divided by 15. So, substituting in these values for our Riemann sum, this is going to be our remand sum is going to be equal to the quantity of 4 divided by 7, plus 4 divided by 9, plus 4 divided by 11. Plus 4 divided by 13, plus 4 divided by 15 in quantity multiplied by delta X, which in our case is just 1. So, evaluating this expression using our calculators, we see that our remod sum is going to be equal to 1.95. And this here is the answer that we're looking for for this problem. So I hope that this explanation has been useful, and I'll see you in the next video.

Midpoint Riemann sums Complete the following steps for the given function, interval, and value of n.

ƒ(𝓍) = 1/x on [1,6] ; n = 5

(d) Calculate the midpoint Riemann sum.

Key Concepts

Riemann Sums

Midpoint Rule

Definite Integral

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