Calculating the Right Riemann Sum for the Integral ∫(831+x)^(1/2)dx using Divide and Multiply Method

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

To find a right Riemann sum for the integral ∫(831+x)^(1/2)dx, we need to divide the interval of integration [8, 31] into subintervals, evaluate the function at the right endpoints of each subinterval, and multiply each value by the width of the subinterval

To find a right Riemann sum for the integral ∫(831+x)^(1/2)dx, we need to divide the interval of integration [8, 31] into subintervals, evaluate the function at the right endpoints of each subinterval, and multiply each value by the width of the subinterval.

Let’s start by determining the number of subintervals. We can use the right Riemann sum formula:

Δx = (b – a) / n

where Δx is the width of each subinterval, a is the lower limit of integration, b is the upper limit of integration, and n is the number of subintervals.

In this case, a = 8 and b = 31, so we have:

Δx = (31 – 8) / n

Next, we need to find the right endpoints of each subinterval. Since the right Riemann sum uses the function value at the right endpoint, we can use the formula:

xi = a + i * Δx, where i = 1, 2, 3, …, n

Let’s say we have n subintervals. Then, the right endpoints will be:

x1 = 8 + Δx = 8 + ((31 – 8) / n)
x2 = x1 + Δx = 8 + 2 * Δx
x3 = x2 + Δx = 8 + 3 * Δx
…
xn = x(n-1) + Δx = 8 + n * Δx

Finally, we multiply each value of f(xi) = (831 + xi)^(1/2) by the width Δx and sum them up:

Right Riemann sum = Δx * [f(x1) + f(x2) + f(x3) + … + f(xn)]

Note that as n approaches infinity, the Riemann sum approaches the actual value of the integral.

To summarize, the right Riemann sum for ∫(831+x)^(1/2)dx is given by:

Right Riemann sum = Δx * [(831 + x1)^(1/2) + (831 + x2)^(1/2) + (831 + x3)^(1/2) + … + (831 + xn)^(1/2)]

where Δx = (31 – 8) / n, x1 = 8 + Δx, x2 = x1 + Δx, x3 = x2 + Δx, …, xn = 8 + n * Δx.

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