Answer
For n = 2, A = 0.584 Units square
For n = 5, A = 0.646 Units square
For n = 10, A = 0.669 Units square
Work Step by Step
1. First, consider n = 2 rectangles.
This means we divide the given interval [1, 2] by 2. Thus, each subinterval has a length of $\frac{1}{n}$ =$\frac{1}{2}$.
The endpoints of the subintervals:
1) f(1+$\frac{1}{2}$) = $\frac{2}{3}$ = 0.667
2) f(2) = $\frac{1}{2}$ = 0.5
Finally, the total area of n = 2 rectangles:
A = $A_{1}$ + $A_{2}$ = $\frac{1}{2}$ $\times$ (0.667 + 0.5) = 0.584 Units square
2. In the same way, we can consider n = 5 rectangles.
This means we divide the given interval [1, 2] by 5. Thus, each subinterval has a length of $\frac{1}{n}$ =$\frac{1}{5}$.
The endpoints of the subintervals:
1) f(1+$\frac{1}{5}$) = $\frac{5}{6}$ = 0.833
2) f(1+$\frac{2}{5}$) = $\frac{5}{7}$ = 0.714
3) f(1+$\frac{3}{5}$) = $\frac{5}{8}$ = 0.625
4) f(1+$\frac{4}{5}$) = $\frac{5}{9}$ = 0.556
5) f(2) = $\frac{1}{2}$ = 0.5
Finally, the total area of n = 5 rectangles:
A = $A_{1}$ + $A_{2}$ + $A_{3}$ + $A_{4}$ + $A_{5}$ = $\frac{1}{5}$ $\times$ (0.833 + 0.714 + 0.625 + 0.556 + 0.5) = 0.646 Units square
3. In the same way, we can consider n = 10 rectangles.
This means we divide the given interval [1, 2] by 10. Thus, each subinterval has a length of $\frac{1}{n}$ =$\frac{1}{10}$.
The endpoints of the subintervals:
1) f(1+$\frac{1}{10}$) = $\frac{10}{11}$ = 0.909
2) f(1+$\frac{2}{10}$) = $\frac{5}{6}$ = 0.833
3) f(1+$\frac{3}{10}$) = $\frac{10}{13}$ = 0.769
4) f(1+$\frac{4}{10}$) = $\frac{5}{7}$ = 0.714
5) f(1+$\frac{5}{10}$) = $\frac{2}{3}$ = 0.667
6) f(1+$\frac{6}{10}$) = $\frac{5}{8}$ = 0.625
7) f(1+$\frac{7}{10}$) = $\frac{10}{17}$ = 0.588
8) f(1+$\frac{8}{10}$) = $\frac{5}{9}$ = 0.556
9) f(1+$\frac{9}{10}$) = $\frac{10}{19}$ = 0.526
10) f(2) = $\frac{1}{2}$ = 0.5
Finally, the total area of n = 10 rectangles:
A = $A_{1}$ + $A_{2}$ + $A_{3}$ + $A_{4}$ + $A_{5}$ + $A_{6}$ + $A_{7}$ + $A_{8}$ + $A_{9}$ + $A_{10}$= $\frac{1}{10}$ $\times$ (0.909 + 0.833 + 0.769 + 0.714 + 0.667 + 0.625 + 0.588 + 0.556 + 0.526 + 0.5) = 0.669 Units square
