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Approximate A(1x,1≤x≤5)A\left(\frac{1}{x} , 1 \leq x \leq 5\right) using 88 left endpoint rectangles. What is that sigma notation to show the sum of the areas of the rectangles.

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Q. Approximate A(1x,1≤x≤5)A\left(\frac{1}{x} , 1 \leq x \leq 5\right) using 88 left endpoint rectangles. What is that sigma notation to show the sum of the areas of the rectangles.
  1. Define function and interval: Step 11: Define the function and interval.\newlineWe need to approximate the area under the curve of f(x)=1xf(x) = \frac{1}{x} from x=1x = 1 to x=5x = 5 using 88 rectangles. The rectangles are based on the left endpoints.
  2. Calculate width of rectangles: Step 22: Calculate the width of each rectangle.\newlineThe total interval length is 5−1=45 - 1 = 4. With 88 rectangles, the width (Δx)(\Delta x) of each rectangle is 48=0.5\frac{4}{8} = 0.5.
  3. Identify left endpoints: Step 33: Identify the xx-coordinates of the left endpoints.\newlineThe left endpoints are x=1x = 1, 1.51.5, 22, 2.52.5, 33, 3.53.5, 44, and 4.54.5.
  4. Calculate heights of rectangles: Step 44: Calculate the heights of the rectangles using f(x)=1xf(x) = \frac{1}{x}. Heights are f(1)f(1), f(1.5)f(1.5), f(2)f(2), f(2.5)f(2.5), f(3)f(3), f(3.5)f(3.5), f(4)f(4), f(4.5)f(4.5).
  5. Write sum of areas in sigma notation: Step 55: Write the sum of the areas of the rectangles using sigma notation.\newlineThe area of each rectangle is height * width = f(xi)∗Δxf(x_i) * \Delta x. The sum of the areas is:\newlineΣi=18\Sigma_{i=1}^{8} of f(xi)∗Δx=Σi=18f(x_i) * \Delta x = \Sigma_{i=1}^{8} of (1xi)∗0.5(\frac{1}{x_i}) * 0.5.
  6. Calculate approximate area: Step 66: Calculate the approximate area.\newlineApproximate area = 0.5×(11+11.5+12+12.5+13+13.5+14+14.5)0.5 \times (\frac{1}{1} + \frac{1}{1.5} + \frac{1}{2} + \frac{1}{2.5} + \frac{1}{3} + \frac{1}{3.5} + \frac{1}{4} + \frac{1}{4.5}).

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