Let f be the function defined by f(x)=3ln(x). If three subintervals of equal length are used, what is the value of the trapezoidal sum approximation for int_(2)^(3.5)3ln(x)dx ? Round to the nearest thousandth if necessary. Answer:

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Let  f be the function defined by  f(x)=3ln(x). If three subintervals of equal length are used, what is the value of the trapezoidal sum approximation for  int_(2)^(3.5)3ln(x)dx ? Round to the nearest thousandth if necessary. Answer:

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Rephrase Problem: Let's first rephrase the  ＂What is the trapezoidal sum approximation for the integral of 3ln(x) from 2 to 3.5 using three subintervals of equal length?＂Divide Interval: To use the trapezoidal rule, we need to divide the interval [2, 3.5] into three equal subintervals. The length of each subinterval is (3.5 - 2) / 3 = 0.5.Evaluate Function: The endpoints of the subintervals are x_0 = 2, x_1 = 2.5, x_2 = 3, and x_3 = 3.5. We will evaluate the function f(x) = 3ln(x) at these points.Apply Trapezoidal Rule: Calculate the function values: f(x_0) = 3ln(2), f(x_1) = 3ln(2.5), f(x_2) = 3ln(3), and f(x_3) = 3ln(3.5).Substitute Values: Now, we apply the trapezoidal rule, which is given by the formula: T = (Δx/2) * [f(x_0) + 2f(x_1) + 2f(x_2) + f(x_3)], where Δx is the length of each subinterval.Perform Calculations: Substitute the values into the trapezoidal rule formula: T = (0.5/2) * [3ln(2) + 2*3ln(2.5) + 2*3ln(3) + 3ln(3.5)].Calculate Sum: Perform the calculations: T = 0.25 * [3ln(2) + 6ln(2.5) + 6ln(3) + 3ln(3.5)].Add Values: Use a calculator to find the numerical values of the logarithms and perform the multiplication: T ≈ 0.25 * [3*0.6931 + 6*0.9163 + 6*1.0986 + 3*1.2528].Multiply for Approximation: Calculate the sum: T ≈ 0.25 * [2.0793 + 5.4978 + 6.5916 + 3.7584].Round to Nearest Thousandth: Add the values together: T ≈ 0.25 * [17.9271].Multiply by 0.25 to get the final approximation: T ≈ 0.25 * 17.9271 ≈ 4.4818.Round to the nearest thousandth: T ≈ 4.482.

Let $f$ be the function defined by $f(x)=3 \ln (x)$ . If three subintervals of equal length are used, what is the value of the trapezoidal sum approximation for $\int_{2}^{3.5} 3 \ln (x) d x$ ? Round to the nearest thousandth if necessary. $\newline$ Answer:

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