Left and right Riemann sums Complete the following steps for t...

Question

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

ƒ(𝓍) = x² ─ 1 on [2,4]; n = 4

(d) Calculate the left and right Riemann sums.

Accepted Answer

Welcome back, everyone. Calculate the right Riemann sum for F of X equals X2 + 2 on the interval from 1 to 3 inclusive with N equals 4 sub-intervals. The first step in this problem is to identify the width of each subinterval delta x. Let's recall that we take the upper bound, which is B, we subtract the lower bound of our interval A, and we divide by N. Our upper bound is 3. We subtract the lower bound, which is 1, and we divide by 4 subintervals. So we get 2 divided by 4, which is 1/2. And now we are going to identify the right endpoints. Starting with the first right endpoint, it is going to be equal to A plus delta X, right? Because we're looking for the right rimann sum. So that'd be 1 + 1/2, and this is equal to 3 halves. Our 2nd value will be equal to. The previous value, which is 3 halves, right? We can label it as X1, and we're going to add delta X. This will be equal to 3 halves plus 1/2, and we get 4 divided by 2, which is 2. X3 is going to be equal to X2 + delta X, which is equal to 2 + 1/2, and this is equal to 5 halves. And finally, our fourth endpoint X4 would be equal to X3 + delta X. That's 5 halves plus 1/2, and this is equal to 3. So now we have our end points. What we are going to do is simply apply the formula, we're going to sum. From K equals 1 up to 4, right, we have 4 subintervals. The value of our function at each. Kal Multiplied by Delta X. Now what we can do is simply factor out delta X. Right, because it is constant, it's always 1/2, and in we're going to add our function value at. 1, specifically X1, right? F of X1 plus F of X2 plus F of X3 plus F of X4. So that's our formula that we want to use. Our delta X is 1/2, we can factor it out. In print Cs, we want to evaluate the value of the function at each X, right? X1, X2, X3, and X4. So we can begin with F of X1. Which essentially means that we're substituting 3 halves into the function. So we get 3 halves squared plus 2. This is equal to 17 divided by 4. We then evaluate F of X2. We substitute 2 into F of X, so that would be 22 + 2. We're going to get 6. F of X3 is going to be equal to 5 halves squared plus 2, which is equal to 33 divided by 4, and F of X4 is going to be equal to +32 + 2, which is equal to 11. So important seats we get 17 divided by 4 + 6 + 33 divided by 4 + 11. So we get 1 half in paris, we can find the common denominator, which is 4, rights, we can just write 4. Our numerator contains 17 + 6 multiplied by 4, that's 24 + 33 + 11 multiplied by 444. Let's simplify We get 118 divided by 4 multiplied by 12, which is equal to 118 divided by 8. Dividing both sides by 2, we get 59 divided by 4, which is our final answer for this problem. Thank you for watching.

Left and right Riemann sums Complete the following steps for the given function, interval, and value of n.ƒ(𝓍) = x² ─ 1 on [2,4]; n = 4(d) Calculate the left and right Riemann sums.

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Left and right Riemann sums Complete the following steps for the given function, interval, and value of n.
ƒ(𝓍) = x² ─ 1 on [2,4]; n = 4
(d) Calculate the left and right Riemann sums.