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Let f be the function defined by f(x)=x^(2). If four subintervals of equal length are used, what is the value of the left Riemann sum approximation for int_(1)^(2)x^(2)dx ? Round to the nearest thousandth if necessary. Answer:

Let f f be the function defined by f(x)=x2 f(x)=x^{2} . If four subintervals of equal length are used, what is the value of the left Riemann sum approximation for ∫12x2dx \int_{1}^{2} x^{2} 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)=x2 f(x)=x^{2} . If four subintervals of equal length are used, what is the value of the left Riemann sum approximation for ∫12x2dx \int_{1}^{2} x^{2} d x ? Round to the nearest thousandth if necessary.\newlineAnswer:
  1. Determine Width of Subintervals: Determine the width of each subinterval. Since we are integrating from x=1x = 1 to x=2x = 2 and we are using four subintervals, the width (Δx\Delta x) of each subinterval is (2−1)/4=0.25(2 - 1) / 4 = 0.25.
  2. Identify Left Endpoints: Identify the xx-values for the left endpoints of each subinterval.\newlineThe left endpoints for the subintervals are x=1x = 1, x=1.25x = 1.25, x=1.5x = 1.5, and x=1.75x = 1.75.
  3. Evaluate Function Values: Evaluate the function f(x)=x2f(x) = x^2 at each of the left endpoints.f(1)=12=1f(1) = 1^2 = 1f(1.25)=1.252=1.5625f(1.25) = 1.25^2 = 1.5625f(1.5)=1.52=2.25f(1.5) = 1.5^2 = 2.25f(1.75)=1.752=3.0625f(1.75) = 1.75^2 = 3.0625
  4. Calculate Rectangle Areas: Multiply each function value by the width of the subintervals to find the area of each rectangle.\newlineArea11 = f(1)×Δx=1×0.25=0.25f(1) \times \Delta x = 1 \times 0.25 = 0.25\newlineArea22 = f(1.25)×Δx=1.5625×0.25=0.390625f(1.25) \times \Delta x = 1.5625 \times 0.25 = 0.390625\newlineArea33 = f(1.5)×Δx=2.25×0.25=0.5625f(1.5) \times \Delta x = 2.25 \times 0.25 = 0.5625\newlineArea44 = f(1.75)×Δx=3.0625×0.25=0.765625f(1.75) \times \Delta x = 3.0625 \times 0.25 = 0.765625
  5. Sum Rectangle Areas: Sum the areas of the rectangles to find the left Riemann sum approximation. \newlineLeft Riemann Sum = Area1+Area2+Area3+Area4=0.25+0.390625+0.5625+0.765625Area_1 + Area_2 + Area_3 + Area_4 = 0.25 + 0.390625 + 0.5625 + 0.765625
  6. Calculate Total Sum: Calculate the total sum to find the approximation.\newlineLeft Riemann Sum = 0.25+0.390625+0.5625+0.765625=1.968750.25 + 0.390625 + 0.5625 + 0.765625 = 1.96875

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