Let f be the function defined by f(x)=4sqrtx. If four subintervals of equal length are used, what is the value of the midpoint Riemann sum approximation for int_(2)^(8)4sqrtxdx ? Round to the nearest thousandth if necessary. Answer:

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Let  f be the function defined by  f(x)=4sqrtx. If four subintervals of equal length are used, what is the value of the midpoint Riemann sum approximation for  int_(2)^(8)4sqrtxdx ? Round to the nearest thousandth if necessary. Answer:

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Calculate Subinterval Length: To calculate the midpoint Riemann sum, we first need to determine the length of each subinterval. The interval [2, 8] has a length of 8 - 2 = 6. Since we are using four subintervals, each subinterval will have a length of 6 / 4 = 1.5.Find Midpoints of Subintervals: Next, we need to find the midpoints of each subinterval. The subintervals are [2, 3.5], [3.5, 5], [5, 6.5], and [6.5, 8]. The midpoints of these subintervals are 2.75, 4.25, 5.75, and 7.25, respectively.Evaluate Function at Midpoints: Now we evaluate the function f(x) = 4√x at each midpoint. This gives us the values f(2.75), f(4.25), f(5.75), and f(7.25). We calculate these values as follows: f(2.75) = 4√2.75 f(4.25) = 4√4.25 f(5.75) = 4√5.75 f(7.25) = 4√7.25Perform Function Value Calculations: We perform the calculations for each function value: f(2.75) = 4√2.75 ≈ 4 * 1.658 ≈ 6.632 f(4.25) = 4√4.25 ≈ 4 * 2.062 ≈ 8.248 f(5.75) = 4√5.75 ≈ 4 * 2.397 ≈ 9.588 f(7.25) = 4√7.25 ≈ 4 * 2.692 ≈ 10.768Calculate Midpoint Riemann Sum: The midpoint Riemann sum is the sum of the function values at the midpoints multiplied by the length of each subinterval. So, we have: Midpoint Riemann Sum = (f(2.75) + f(4.25) + f(5.75) + f(7.25)) * length of subinterval Midpoint Riemann Sum ≈ (6.632 + 8.248 + 9.588 + 10.768) * 1.5Calculate Sum of Function Values: We calculate the sum of the function values and then multiply by the length of the subinterval: Midpoint Riemann Sum ≈ (6.632 + 8.248 + 9.588 + 10.768) * 1.5 Midpoint Riemann Sum ≈ 35.236 * 1.5 Midpoint Riemann Sum ≈ 52.854Round Result to Nearest Thousandth: We round the result to the nearest thousandth as requested: Midpoint Riemann Sum ≈ 52.854 (rounded to three decimal places)

Let $f$ be the function defined by $f(x)=4 \sqrt{x}$ . If four subintervals of equal length are used, what is the value of the midpoint Riemann sum approximation for $\int_{2}^{8} 4 \sqrt{x} d x$ ? Round to the nearest thousandth if necessary. $\newline$ Answer:

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