Let f be the function defined by f(x)=5ln(x). If three subintervals of equal length are used, what is the value of the trapezoidal sum approximation for int_(3)^(7.5)5ln(x)dx? Round to the nearest thousandth if necessary. Answer:

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Let  f be the function defined by  f(x)=5ln(x). If three subintervals of equal length are used, what is the value of the trapezoidal sum approximation for  int_(3)^(7.5)5ln(x)dx? Round to the nearest thousandth if necessary. Answer:

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Calculate Width of Subintervals: Determine the width of each subinterval. Since we are using three subintervals of equal length from 3 to 7.5, we calculate the width (h) as follows: h = (7.5 - 3) / 3 h = 4.5 / 3 h = 1.5Identify x-Values for Subintervals: Identify the x-values for the subintervals. The x-values at the endpoints of the subintervals are: x_0 = 3 x_1 = 3 + 1.5 = 4.5 x_2 = 4.5 + 1.5 = 6 x_3 = 6 + 1.5 = 7.5Calculate Function Values: Calculate the function values at the x-values. f(x) = 5ln(x), so we calculate f(x) for each x-value: f(x_0) = 5ln(3) f(x_1) = 5ln(4.5) f(x_2) = 5ln(6) f(x_3) = 5ln(7.5)Apply Trapezoidal Sum Formula: Apply the trapezoidal sum formula. The trapezoidal sum T is given by: T = (h/2) * [f(x_0) + 2f(x_1) + 2f(x_2) + f(x_3)] Substitute the function values and h into the formula: T = (1.5/2) * [5ln(3) + 2*5ln(4.5) + 2*5ln(6) + 5ln(7.5)]Perform Calculations: Perform the calculations. T = (0.75) * [5ln(3) + 10ln(4.5) + 10ln(6) + 5ln(7.5)] T = 0.75 * [15.104412573 + 32.2361913 + 35.67926935 + 20.79441567] T = 0.75 * [103.8142889] T ≈ 77.86071668 Round to the nearest thousandth: T ≈ 77.861

Let $f$ be the function defined by $f(x)=5 \ln (x)$ . If three subintervals of equal length are used, what is the value of the trapezoidal sum approximation for $\int_{3}^{7.5} 5 \ln (x) d x ?$ Round to the nearest thousandth if necessary. $\newline$ Answer:

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