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.

{Use of Tech} ƒ(𝓍) = √x on [1,3] ; n = 4

(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 Riemann sum for F of X is equal to the square root of the quantity of X + 1 in quantity on the closed interval from 0 to 2 with N equal to 4 sub-intervals. So first step is to determine our sub-interval width, which we'll denote as delta X. And we call that our sub-interval width delta X is going to be equal to, and here we'll have our upper bound of our interval, which is going to be 2, minus our lower bound, which is 0, and that quantity is going to be divided by the number of sub-intervals N, which in our case is equal to 4. And when we evaluate this expression, we see that Delta X is going to be equal to 1 divided by 2. Next, we need to determine the midpoints for each of our sub-intervals. So, for our first sub-intervals midpoint, which we'll call X1, recall that this is going to be equal to our lower bound of our interval, which in this case is going to be 0. Plus, 1/2 multiplied by delta X. And since delta X is equal to 1/2, this is going to be equal to 0 plus 1/2 multiplied by 1/2, which, when we evaluate this expression, is going to be equal to 1 divided by 4. Now, to find the midpoint of the next subinterval, we just need to add delta X to the midpoint of our first interval, and that's how we'll find the midpoint of each of our successive subintervals, which is by adding delta X to the previous midpoint. So, for X2, which is going to be the midpoint of the second sub-interval, that's going to be equal to X1, which is going to be 1 divided by 4, plus delta X, which is 1 divided by 2. And if I win this expression, this is going to be equal to 3 divided by 4. For the next midpoint, for the next subinterval, we'll call that X3, and this is going to be equal to X2, which is 3, divided by 4, plus delta X, which is 1 divided by 2, and this is going to be equal to 5 divided by 4. And the midpoint of our last subinterval, which we'll call X4, that's going to be equal to X3, which is 5 divided by 4, plus delta X, which is 1 divided by 2. And if I in this expression, we see that this is going to be equal 2. 7 divided by 4. So now that we have identified the midpoints for each of our sum intervals, we're gonna look at calculating our remod sum. So, we call for the red sum that we'll have the sum. Of K equal to 1 to 4, since N is equal to 4 of F of X of K. 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. And that whole quantity is multiplied by delta X. So we need to do is to evaluate F of X at each of our midpoints. So, for our first midpoint, we'll calculate F of X1. Which is going to be equal to. F of 1 divided by 4. Which is going to be equal to the square root of the quantity of 1 divided by 4 plus 1 in quantity. And valuating this quantity, this is going to be equal to the square root of 5 divided by 2. Next, we need to calculate F of X2. Which is gonna be F of 3 divided by 4. Which is equal to the square root of the quantity of 3 divided by 4, plus 1 in quantity, and evaluating this expression, we see that this is going to be equal to. Square root of 7, divided by 2. Next, we'll need to calculate F of X3, which is going to be equal to F of 5 divided by 4. And that's going to be equal to the square root of the quantity of 5 divided by 4, plus 1 in quantity. And if I weigh in this expression, we see that this is gonna be equal to 3 divided by 2. And finally, for our last midpoint, we'll have F of X4. Which is equal to F of 7 divided by 4. Which is equal to the square root of the quantity of 7 divided by 4, plus 1 in quantity, and if I this this expression, we see that this is going to be equal to the square root of 11 divided by 2. So now we just see substitute in those values into our expression for our remod sum. In doing so, we see that this is going to be equal to the quantity of the square root of 5, divided by 2. Plus the square root of 7 divided by 2, plus 3 divided by 2. Plus, the square root of 11, divided by 2, in quantity multiplied by delta X, which in this case is 1 divided by 2. And so if I weigh in this expression with our calculator, we see that our remand sum is going to be equal to. 2.80. So here we just round it to two decimal places. So 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.

{Use of Tech} ƒ(𝓍) = √x on [1,3] ; n = 4

(d) Calculate the midpoint Riemann sum.

Key Concepts

Riemann Sums

Midpoint Rule

Definite Integral

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