Answer
For n = 2, A = 0.433 Units square
For n = 5, A = 0.659 Units square
For n = 10, A = 0.726 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($\frac{1}{2}$) = $\sqrt {1-\frac{1}{2}^{2}}$ = 0.866
2) f(1) = $\sqrt {1-1^{2}}$ = 0
Finally, the total area of n = 2 rectangles:
A = $A_{1}$ + $A_{2}$ = $\frac{1}{2}$ $\times$ (0.866 + 0) = 0.433 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($\frac{1}{5}$) = $\sqrt {1-\frac{1}{5}^{2}}$ = 0.980
2) f($\frac{2}{5}$) = $\sqrt {1-\frac{2}{5}^{2}}$ = 0.917
3) f($\frac{3}{5}$) = $\sqrt {1-\frac{3}{5}^{2}}$ = 0.8
4) f($\frac{4}{5}$) = $\sqrt {1-\frac{4}{5}^{2}}$ = 0.6
5) f(1) = $\sqrt {1-1^{2}}$ = 0
Finally, the total area of n = 5 rectangles:
A = $A_{1}$ + $A_{2}$ + $A_{3}$ + $A_{4}$ + $A_{5}$ = $\frac{1}{5}$ $\times$ (0.980 + 0.917 + 0.8 + 0.6 + 0) = 0.659 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($\frac{1}{10}$) = $\sqrt {1-\frac{1}{10}^{2}}$ = 0.995
2) f($\frac{2}{10}$) = $\sqrt {1-\frac{1}{5}^{2}}$ = 0.980
3) f($\frac{3}{10}$) = $\sqrt {1-\frac{3}{10}^{2}}$ = 0.954
4) f($\frac{4}{10}$) = $\sqrt {1-\frac{2}{5}^{2}}$ = 0.917
5) f($\frac{5}{10}$) = $\sqrt {1-\frac{1}{2}^{2}}$ = 0.866
6) f($\frac{6}{10}$) = $\sqrt {1-\frac{3}{5}^{2}}$ = 0.8
7) f($\frac{7}{10}$) = $\sqrt {1-\frac{7}{10}^{2}}$ = 0.714
8) f($\frac{8}{10}$) = $\sqrt {1-\frac{4}{5}^{2}}$ = 0.6
9) f($\frac{9}{10}$) = $\sqrt {1-\frac{9}{10}^{2}}$ = 0.436
10) f(1) = $\sqrt {1-1^{2}}$ = 0
Finally, the total area of n = 10 rectangles:
A = $A_{1}$ + $A_{2}$ + $A_{3}$ + $A_{4}$ + $A_{5}$ + $A_{6}$ + $A_{7}$ + $A_{8}$ + $A_{9}$ + $A_{10}$= $\frac{1}{10}$ $\times$ (0.995 + 0.980 + 0.954 + 0.917 + 0.866 + 0.8 + 0.714 + 0.6 + 0.436 + 0) = 0.726 Units square