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Let f be the function defined by f(x)=6sqrtx. If three subintervals of equal length are used, what is the value of the right Riemann sum approximation for int_(3)^(7.5)6sqrtxdx ? Round to the nearest thousandth if necessary. Answer:

Let f f be the function defined by f(x)=6x f(x)=6 \sqrt{x} . If three subintervals of equal length are used, what is the value of the right Riemann sum approximation for ∫37.56xdx \int_{3}^{7.5} 6 \sqrt{x} d x ? Round to the nearest thousandth if necessary.\newlineAnswer:

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Q. Let f f be the function defined by f(x)=6x f(x)=6 \sqrt{x} . If three subintervals of equal length are used, what is the value of the right Riemann sum approximation for ∫37.56xdx \int_{3}^{7.5} 6 \sqrt{x} d x ? Round to the nearest thousandth if necessary.\newlineAnswer:
  1. Divide into Subintervals: To approximate the integral using a right Riemann sum, we first need to divide the interval [3,7.5][3, 7.5] into three subintervals of equal length. The length of each subinterval is calculated by subtracting the lower bound from the upper bound and dividing by the number of subintervals.\newlineLength of each subinterval = (7.5−3)/3=4.5/3=1.5(7.5 - 3) / 3 = 4.5 / 3 = 1.5
  2. Find Right Endpoints: Next, we need to find the right endpoints of each subinterval. Since we start at x=3x = 3 and each subinterval has a length of 1.51.5, the right endpoints will be at x=3+1.5x = 3 + 1.5, x=3+2(1.5)x = 3 + 2(1.5), and x=3+3(1.5)x = 3 + 3(1.5).
    Right endpoints: x1=4.5x_1 = 4.5, x2=6x_2 = 6, x3=7.5x_3 = 7.5
  3. Evaluate Function: Now we evaluate the function f(x)=6xf(x) = 6\sqrt{x} at each of the right endpoints.f(x1)=64.5f(x_1) = 6\sqrt{4.5}f(x2)=66f(x_2) = 6\sqrt{6}f(x3)=67.5f(x_3) = 6\sqrt{7.5}
  4. Calculate Function Values: We calculate the values of the function at these points. \newlinef(x1)=64.5≈6×2.121=12.726f(x_1) = 6\sqrt{4.5} \approx 6 \times 2.121 = 12.726\newlinef(x2)=66≈6×2.449=14.694f(x_2) = 6\sqrt{6} \approx 6 \times 2.449 = 14.694\newlinef(x3)=67.5≈6×2.738=16.428f(x_3) = 6\sqrt{7.5} \approx 6 \times 2.738 = 16.428
  5. Calculate Riemann Sum: The right Riemann sum is the sum of the products of the function values at the right endpoints and the length of each subinterval.\newlineRight Riemann sum = f(x1)×length+f(x2)×length+f(x3)×lengthf(x_1) \times \text{length} + f(x_2) \times \text{length} + f(x_3) \times \text{length}\newlineRight Riemann sum ≈12.726×1.5+14.694×1.5+16.428×1.5\approx 12.726 \times 1.5 + 14.694 \times 1.5 + 16.428 \times 1.5
  6. Perform Calculations: We perform the calculations to find the approximate value of the integral.\newlineRight Riemann sum ≈12.726×1.5+14.694×1.5+16.428×1.5\approx 12.726 \times 1.5 + 14.694 \times 1.5 + 16.428 \times 1.5\newlineRight Riemann sum ≈19.089+22.041+24.642\approx 19.089 + 22.041 + 24.642\newlineRight Riemann sum ≈65.772\approx 65.772
  7. Round the Result: We round the result to the nearest thousandth as instructed.\newlineRight Riemann sum ≈65.772\approx 65.772 (rounded to the nearest thousandth)

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