Let f be the function defined by f(x)=(2)/(x). If three subintervals of equal length are used, what is the value of the trapezoidal sum approximation for int_(3)^(9)(2)/(x)dx ? Round to the nearest thousandth if necessary. Answer:

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Let  f be the function defined by  f(x)=(2)/(x). If three subintervals of equal length are used, what is the value of the trapezoidal sum approximation for  int_(3)^(9)(2)/(x)dx ? Round to the nearest thousandth if necessary. Answer:

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Calculate Subinterval Width: To approximate the integral of f(x) = 2/x from x = 3 to x = 9 using the trapezoidal rule with three subintervals, we first need to calculate the width of each subinterval. The total interval length is 9 - 3 = 6. Since we are using three subintervals, the width (h) of each subinterval is 6/3 = 2.Evaluate Function at Endpoints: Next, we need to evaluate the function f(x) at the endpoints of the subintervals. The endpoints are x = 3, x = 5, x = 7, and x = 9. We calculate f(3), f(5), f(7), and f(9). f(3) = 2/3 f(5) = 2/5 f(7) = 2/7 f(9) = 2/9Apply Trapezoidal Rule: Now we apply the trapezoidal rule, which is given by the formula: Trapezoidal sum = (h/2) * [f(x_0) + 2*f(x_1) + 2*f(x_2) + ... + 2*f(x_{n-1}) + f(x_n)] where h is the width of each subinterval, x_0 to x_n are the endpoints of the subintervals, and n is the number of subintervals.Perform Addition: Plugging in the values we have: Trapezoidal sum = (2/2) * [f(3) + 2*f(5) + 2*f(7) + f(9)] = 1 * [(2/3) + 2*(2/5) + 2*(2/7) + (2/9)] = (2/3) + (4/5) + (4/7) + (2/9)Convert to Common Denominator: We now perform the addition: = (2/3) + (4/5) + (4/7) + (2/9) To add these fractions, we need a common denominator, which is the least common multiple of 3, 5, 7, and 9. The LCM of these numbers is 315. So we convert each fraction to have the denominator of 315: = (2/3)*(105/105) + (4/5)*(63/63) + (4/7)*(45/45) + (2/9)*(35/35) = (210/315) + (252/315) + (180/315) + (70/315)Add Fractions: Adding the fractions together, we get: = (210 + 252 + 180 + 70) / 315 = 712 / 315 Now we simplify the fraction if possible and round to the nearest thousandth. 712 / 315 ≈ 2.260

Let $f$ be the function defined by $f(x)=\frac{2}{x}$ . If three subintervals of equal length are used, what is the value of the trapezoidal sum approximation for $\int_{3}^{9} \frac{2}{x} d x$ ? Round to the nearest thousandth if necessary. $\newline$ Answer:

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Let $f$ be the function defined by $f(x)=\frac{2}{x}$ . If three subintervals of equal length are used, what is the value of the trapezoidal sum approximation for $\int_{3}^{9} \frac{2}{x} d x$ ? Round to the nearest thousandth if necessary. $\newline$ Answer: