Use the Midpoint Rule with the given value of n to approximate the integral. Round the answer to four decimal places.

int_2^10 2 sqrt(x^2+5)dx text(, ) n=4

To approximate the integral using the Midpoint Rule with n = 4, we need to divide the interval [2, 10] into 4 subintervals of equal width.

First, let's find the width of each subinterval:

Width (Δx) = (10 - 2) / n = (10 - 2) / 4 = 8 / 4 = 2

So, each subinterval has a width of 2.

Next, we'll find the midpoints of these subintervals. Since we have 4 subintervals, there will be 5 points: the endpoints of the interval and the 3 midpoints.

The midpoints are given by:

x₁ = 2 + (Δx / 2) = 2 + (2 / 2) = 2 + 1 = 3
x₂ = 2 + (3 * Δx / 2) = 2 + (3 * 2 / 2) = 2 + 3 = 5
x₃ = 2 + (5 * Δx / 2) = 2 + (5 * 2 / 2) = 2 + 5 = 7
x₄ = 2 + (7 * Δx / 2) = 2 + (7 * 2 / 2) = 2 + 7 = 9
x₅ = 10

Now, we find the value of the function at these midpoints, which in this case is 2 * sqrt(x² + 5).

f(x₁) = 2 * sqrt(3² + 5) = 2 * sqrt(9 + 5) = 2 * sqrt(14)
f(x₂) = 2 * sqrt(5² + 5) = 2 * sqrt(25 + 5) = 2 * sqrt(30)
f(x₃) = 2 * sqrt(7² + 5) = 2 * sqrt(49 + 5) = 2 * sqrt(54)
f(x₄) = 2 * sqrt(9² + 5) = 2 * sqrt(81 + 5) = 2 * sqrt(86)
f(x₅) = 2 * sqrt(10² + 5) = 2 * sqrt(100 + 5) = 2 * sqrt(105)

Now, we can compute the approximate area using the Midpoint Rule formula:

Approximate Area ≈ Δx * (f(x₁) + f(x₂) + f(x₃) + f(x₄))

≈ 2 * (f(x₁) + f(x₂) + f(x₃) + f(x₄))
≈ 2 * (f(3) + f(5) + f(7) + f(9))

Now, plug in the values we calculated earlier to find the approximate area:

≈ 2 * (2 * sqrt(14) + 2 * sqrt(30) + 2 * sqrt(54) + 2 * sqrt(86))

Finally, calculate this expression using a calculator or software and round the answer to four decimal places.