Optimizing Right Riemann Sum Calculation For ∫831+X−−−−−√ⅆx Using N=4

Which of the following is a right Riemann sum for ∫831+x−−−−−√ⅆx ?

∑k=1n(4+5kn−−−−−−√⋅5n) – has the 4 + quantity and 5/n on the right side

To find the right Riemann sum for the given integral ∫831+x−−−−−√ⅆx, we need to use the right endpoints of subintervals for evaluation.

Let’s divide the interval [8, 13] into n subintervals of equal width:

Δx = (13 – 8) / n = 5 / n

The right endpoint of the kth subinterval is given by:

xk = 8 + kΔx

The height of the rectangle corresponding to the kth subinterval is given by:

f(xk) = (8 + xk)^(1/2)

The right Riemann sum for the integral is given by:

R_n = ∑[k=1 to n] f(xk)Δx

R_n = ∑[k=1 to n] (8 + xk)^(1/2) (5 / n)

R_n = [(8 + x1)^(1/2) + (8 + x2)^(1/2) + … + (8 + xn)^(1/2)] (5 / n)

Since we are asked to provide one of the right Riemann sums, we need to choose a value of n and calculate the corresponding R_n. Let’s choose n = 4:

Δx = 5 / 4 = 1.25

x1 = 8 + 1.25 = 9.25

x2 = 8 + 2(1.25) = 10.5

x3 = 8 + 3(1.25) = 11.75

x4 = 8 + 4(1.25) = 13

f(x1) = (8 + 9.25)^(1/2) ≈ 3.683

f(x2) = (8 + 10.5)^(1/2) ≈ 3.969

f(x3) = (8 + 11.75)^(1/2) ≈ 4.228

f(x4) = (8 + 13)^(1/2) ≈ 4.472

R_4 = [(8 + 9.25)^(1/2) + (8 + 10.5)^(1/2) + (8 + 11.75)^(1/2) + (8 + 13)^(1/2)] (5 / 4)

R_4 ≈ 18.346

Therefore, the right Riemann sum for ∫831+x−−−−−√ⅆx using n=4 is approximately 18.346.

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