Use the Midpoint Rule with the given value of n to approximate the integral. Round the answer to four decimal places.𝜋7x sin2(x) dx,0      n = 4

The Midpoint Rule is a numerical method to approximate the definite integral. The formula for the Midpoint Rule is:

∫(from a to b) f(x) dx ≈ Δx [f(x1) + f(x2) + ... + f(xn)]

where Δx = (b - a) / n and xi = a + Δx/2 + iΔx for i = 0, 1, ..., n-1.

Given the integral ∫(from 0 to π) 7x sin^2(x) dx with n = 4, we first calculate Δx:

Δx = (π - 0) / 4 = π/4.

Then we calculate the xi values:

x0 = 0 + π/8 = π/8, x1 = π/8 + π/4 = 3π/8, x2 = 3π/8 + π/4 = 5π/8, x3 = 5π/8 + π/4 = 7π/8.

Next, we substitute these xi values into the function f(x) = 7x sin^2(x):

f(x0) = 7(π/8) sin^2(π/8), f(x1) = 7(3π/8) sin^2(3π/8), f(x2) = 7(5π/8) sin^2(5π/8), f(x3) = 7(7π/8) sin^2(7π/8).

Finally, we substitute these values into the Midpoint Rule formula:

∫(from 0 to π) 7x sin^2(x) dx ≈ π/4 [f(x0) + f(x1) + f(x2) + f(x3)].

After calculating the right-hand side, round the result to four decimal places to get the final answer.