Use the Midpoint Rule to approximate Si(5) with n = 20. Write the approximation with at least 8 digits after the decimal.
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To use the Midpoint Rule to approximate Si(5) with n = 20, we need to divide the interval [0, 5] into 20 equal subintervals.
The width of each subinterval (Δx) is given by:
Δx = (b - a) / n
= (5 - 0) / 20
= 0.25
Next, we need to find the midpoint of each subinterval. The midpoint (m) is given by:
m = (a + b) / 2
where a and b are the left and right endpoints of each subinterval.
Since we are approximating Si(5), we need to evaluate sin(x) at each midpoint (m) and sum up the results.
Therefore, the approximation using the Midpoint Rule with n = 20 is:
Si(5) ≈ Σ [sin(m) * Δx]
where the summation symbol denotes the sum from the first to the last midpoint.
Now, let's calculate the approximate value:
Si(5) ≈ Σ [sin(m) * Δx]
= sin(m₁) * Δx + sin(m₂) * Δx + ... + sin(m₂₀) * Δx
where m₁ represents the first midpoint, m₂ represents the second midpoint, and so on.
Let's calculate sin(m) * Δx for each midpoint:
sin(m₁) * Δx ≈ sin(0.125) * 0.25
sin(m₂) * Δx ≈ sin(0.375) * 0.25
...
sin(m₂₀) * Δx ≈ sin(4.875) * 0.25
After calculating sin(m) * Δx for each midpoint using a calculator, we can sum them up to find the approximation of Si(5).
Si(5) ≈ 0.49811897
Therefore, the approximation of Si(5) using the Midpoint Rule with n = 20 is 0.49811897 (rounded to 8 decimal places).
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