using rectangles each of whose height is given by the value of the function at the mid - point of the…

using rectangles each of whose height is given by the value of the function at the mid - point of the rectangles base (the mid - point rule), estimate the area under the graph of the following function, using first two and then four rectangles. f(x)=2/x between x = 1/9 and x = 1. using two rectangles, the estimate for the area under the curve is (type an exact answer.)

using rectangles each of whose height is given by the value of the function at the mid - point of the rectangles base (the mid - point rule), estimate the area under the graph of the following function, using first two and then four rectangles. f(x)=2/x between x = 1/9 and x = 1. using two rectangles, the estimate for the area under the curve is (type an exact answer.)

Answer

Explanation:

Step1: Calculate width of rectangles for (n = 2)

The interval is ([a=\frac{1}{9},b = 1]). For (n = 2), the width (\Delta x=\frac{b - a}{n}=\frac{1-\frac{1}{9}}{2}=\frac{\frac{8}{9}}{2}=\frac{4}{9}).

Step2: Find mid - points

The mid - points of the two sub - intervals are (x_1=\frac{\frac{1}{9}+\frac{1}{9}+\frac{4}{9}}{2}=\frac{\frac{6}{9}}{2}=\frac{1}{3}) and (x_2=\frac{\frac{1}{9}+\frac{4}{9}+\ 1}{2}=\frac{\frac{14}{9}}{2}=\frac{7}{9}).

Step3: Evaluate the function at mid - points

(f(x)=\frac{2}{x}), so (f(x_1)=\frac{2}{\frac{1}{3}} = 6) and (f(x_2)=\frac{2}{\frac{7}{9}}=\frac{18}{7}).

Step4: Calculate the area estimate

The area estimate (A=\sum_{i = 1}^{2}f(x_i)\Delta x=\Delta x(f(x_1)+f(x_2))=\frac{4}{9}(6+\frac{18}{7})=\frac{4}{9}\times\frac{42 + 18}{7}=\frac{4}{9}\times\frac{60}{7}=\frac{80}{21}).

Answer:

(\frac{80}{21})