Use the Midpoint Rule with the given value of n to approximate the integral. Round the answer to four decimal places. 880sin(x) dx, n = 4
Question
Use the Midpoint Rule with the given value of n to approximate the integral. Round the answer to four decimal places. 880sin(x) dx, n = 4
Solution
The Midpoint Rule is a numerical method to approximate the definite integral of a function. The formula for the Midpoint Rule is:
∫_a^b f(x) dx ≈ Δx [f(x_1) + f(x_2) + ... + f(x_n)]
where Δx = (b - a) / n and x_i = a + Δx/2 + iΔx for i = 0, 1, ..., n-1.
In this case, we are given that the function f(x) = 880sin(x), the interval is [0, 1], and n = 4. So, we first need to calculate Δx:
Δx = (1 - 0) / 4 = 0.25
Next, we calculate the x_i values and the corresponding f(x_i) values:
x_0 = 0 + 0.25/2 = 0.125, f(x_0) = 880sin(0.125) x_1 = 0.125 + 0.25 = 0.375, f(x_1) = 880sin(0.375) x_2 = 0.375 + 0.25 = 0.625, f(x_2) = 880sin(0.625) x_3 = 0.625 + 0.25 = 0.875, f(x_3) = 880sin(0.875)
Finally, we substitute these values into the Midpoint Rule formula:
∫_0^1 880sin(x) dx ≈ 0.25 [880sin(0.125) + 880sin(0.375) + 880sin(0.625) + 880sin(0.875)]
Calculate the right-hand side of this equation to get the approximate value of the integral. Remember to round your answer to four decimal places.
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