Answer
For n = 2, A = 0.854 Units square
For n = 5, A = 0.750 Units square
For n = 10, A = 0.711 Units square
Work Step by Step
1. First, consider n = 2 rectangles.
This means we divide the given interval [0, 1] by 2. Thus, each subinterval has a length of $\frac{1}{n}$ = $\frac{1}{2}$.
The endpoints of the subintervals:
1) f($\frac{1}{2}$) = $\frac{1}{\sqrt 2}$ = 0.707
2) f(1) = $\sqrt 1$ = 1
Finally, the total area of n = 2 rectangles:
A = $A_{1}$ + $A_{2}$ = $\frac{1}{2}$ $\times$ (0.707 + 1) = 0.854 Units square
2. In the same way, we can consider n = 5 rectangles.
This means we divide the given interval [0, 1] by 5. Thus, each subinterval has a length of $\frac{1}{n}$ = $\frac{1}{5}$.
The endpoints of the subintervals:
1) f($\frac{1}{5}$) = $\frac{1}{\sqrt 5}$ = 0.447
2) f($\frac{2}{5}$) = $\sqrt {\frac{2}{5}}$ = 0.632
3) f($\frac{3}{5}$) = $\sqrt {\frac{3}{5}}$ = 0.775
4) f($\frac{4}{5}$) = $\sqrt {\frac{4}{5}}$ = 0.894
5) f(1) = $\sqrt 1$ = 1
Finally, the total area of n = 5 rectangles:
A = $A_{1}$ + $A_{2}$ + $A_{3}$ + $A_{4}$ + $A_{5}$ = $\frac{1}{5}$ $\times$ (0.447 + 0.632 + 0.775 + 0.894 + 1) = 0.750 Units square
3. In the same way, we can consider n = 10 rectangles.
This means we divide the given interval [0, 1] by 10. Thus, each subinterval has a length of $\frac{1}{n}$ = $\frac{1}{10}$.
The endpoints of the subintervals:
1) f($\frac{1}{10}$) = $\frac{1}{\sqrt 10}$ = 0.316
2) f($\frac{2}{10}$) = $\sqrt {\frac{1}{5}}$ = 0.447
3) f($\frac{3}{10}$) = $\sqrt {\frac{3}{10}}$ = 0.548
4) f($\frac{4}{10}$) = $\sqrt {\frac{2}{5}}$ = 0.632
5) f($\frac{5}{10}$) = $\sqrt {\frac{1}{2}}$ = 0.707
6) f($\frac{6}{10}$) = $\sqrt {\frac{3}{5}}$ = 0.775
7) f($\frac{7}{10}$) = $\sqrt {\frac{7}{10}}$ = 0.837
8) f($\frac{8}{10}$) = $\sqrt {\frac{4}{5}}$ = 0.894
9) f($\frac{9}{10}$) = $\sqrt {\frac{9}{10}}$ = 0.949
10) f(1) = $\sqrt 1$ = 1
Finally, the total area of n = 10 rectangles:
A = $A_{1}$ + $A_{2}$ + $A_{3}$ + $A_{4}$ + $A_{5}$ + $A_{6}$ + $A_{7}$ + $A_{8}$ + $A_{9}$ + $A_{10}$= $\frac{1}{10}$ $\times$ (0.316 + 0.447 + 0.548 + 0.632 + 0.707 + 0.775 + 0.837 + 0.894 + 0.949 + 1) = 0.711 Units square
