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 left Riemann sum approximation for int_(1)^(10)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 left Riemann sum approximation for  int_(1)^(10)3ln(x)dx ? Round to the nearest thousandth if necessary. Answer:

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Calculate Width of Subintervals: To calculate the left Riemann sum, we first need to determine the width of each subinterval. The interval [1, 10] has a length of 10 - 1 = 9. Since we are using three subintervals of equal length, each subinterval will have a width of 9 / 3 = 3.Find X-Values for Left Riemann Sum: Next, we need to find the x-values at which we will evaluate the function f(x) = 3ln(x) for the left Riemann sum. These x-values are the left endpoints of each subinterval. The first subinterval is [1, 4], the second is [4, 7], and the third is [7, 10]. Therefore, the x-values are 1, 4, and 7.Evaluate Function at X-Values: Now we evaluate the function f(x) at each of these x-values. f(1) = 3ln(1) = 0, f(4) = 3ln(4), and f(7) = 3ln(7).Calculate Left Riemann Sum: We can now calculate the left Riemann sum by multiplying each function value by the width of the subintervals and summing them up. The left Riemann sum is (f(1) * 3) + (f(4) * 3) + (f(7) * 3).Substitute Function Values: Substitute the function values into the sum to get the approximation: (0 * 3) + (3ln(4) * 3) + (3ln(7) * 3).Perform Multiplication: Perform the multiplication: 0 + (3 * 3 * ln(4)) + (3 * 3 * ln(7)).Calculate Numerical Values: Calculate the numerical values: 0 + (9 * ln(4)) + (9 * ln(7)).Calculate Approximation: Using a calculator, we find that ln(4) ≈ 1.386 and ln(7) ≈ 1.946. So the approximation becomes 0 + (9 * 1.386) + (9 * 1.946).Perform Final Calculations: Now we perform the calculations: 0 + (9 * 1.386) + (9 * 1.946) = 0 + 12.474 + 17.514.Round to Nearest Thousandth: Adding these values together gives us the left Riemann sum approximation: 12.474 + 17.514 = 29.988.Round to the nearest thousandth if necessary. The approximation is already at the thousandth place, so rounding is not needed. The final answer is 29.988.

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 left Riemann sum approximation for $\int_{1}^{10} 3 \ln (x) d x$ ? Round to the nearest thousandth if necessary. $\newline$ Answer:

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