Answer
For n = 2, A = 1 Units square
For n = 5, A = 1.424 Units square
For n = 10, A = 1.519 Units square
Work Step by Step
1. First, consider n = 2 rectangles.
This means we divide the given interval [-1, 1] by 2. Thus, each subinterval has a length of $\frac{2}{n}$=$\frac{2}{2}$=1.
The endpoints of the subintervals:
1) f(-1+1) = $\sqrt 1$ = 1
2) f(1) = $\sqrt {1-1^{2}}$ = 0
Finally, the total area of n = 2 rectangles:
A = $A_{1}$ + $A_{2}$ = 1 $\times$ (1 + 0) = 1 Units square
2. In the same way, we can consider n = 5 rectangles.
This means we divide the given interval [-1, 1] by 5. Thus, each subinterval has a length of $\frac{2}{n}$=$\frac{2}{5}$.
The endpoints of the subintervals:
1) f(-1+$\frac{2}{5}$) = f(-$\frac{3}{5}$) = $\sqrt {1-(-\frac{3}{5})^{2}}$ = 0.8
2) f(-1+$\frac{4}{5}$) = f(-$\frac{1}{5}$) = $\sqrt {1-(-\frac{1}{5})^{2}}$ = 0.98
3) f(-1+$\frac{6}{5}$) = f($\frac{1}{5}$) = $\sqrt {1-\frac{1}{5}^{2}}$ = 0.98
4) f(-1+$\frac{8}{5}$) = f($\frac{3}{5}$) = $\sqrt {1-\frac{3}{5}^{2}}$ = 0.8
5) f(1) = $\sqrt {1-1^{2}}$ = 0
Finally, the total area of n = 5 rectangles:
A = $A_{1}$ + $A_{2}$ + $A_{3}$ + $A_{4}$ + $A_{5}$ = $\frac{2}{5}$ $\times$ (0.8 + 0.98 + 0.98 + 0.8 + 0) = 1.424 Units square
3. In the same way, we can consider n = 10 rectangles.
This means we divide the given interval [-1, 1] by 10. Thus, each subinterval has a length of $\frac{2}{n}$=$\frac{2}{10}$=$\frac{1}{5}$.
The endpoints of the subintervals:
1) f(-1+$\frac{1}{5}$) = f(-$\frac{4}{5}$) = $\sqrt {1-(-\frac{4}{5})^{2}}$ = 0.6
2) f(-1+$\frac{2}{5}$) = f(-$\frac{3}{5}$) = $\sqrt {1-(-\frac{3}{5})^{2}}$ = 0.8
3) f(-1+$\frac{3}{5}$) = f(-$\frac{2}{5}$) = $\sqrt {1-(-\frac{2}{5})^{2}}$ = 0.917
4) f(-1+$\frac{4}{5}$) = f(-$\frac{1}{5}$) = $\sqrt {1-(-\frac{1}{5})^{2}}$ = 0.98
5) f(-1+1) = $\sqrt 1$ = 1
6) f(-1+$\frac{6}{5}$) = f($\frac{1}{5}$) = $\sqrt {1-\frac{1}{5}^{2}}$ = 0.98
7) f(-1+$\frac{7}{5}$) = f($\frac{2}{5}$) = $\sqrt {1-\frac{2}{5}^{2}}$ = 0.917
8) f(-1+$\frac{8}{5}$) = f($\frac{3}{5}$) = $\sqrt {1-\frac{3}{5}^{2}}$ = 0.8
9) f(-1+$\frac{9}{5}$) = f($\frac{4}{5}$) = $\sqrt {1-\frac{4}{5}^{2}}$ = 0.6
10) f(1) = $\sqrt {1-1^{2}}$ = 0
Finally, the total area of n = 10 rectangles:
A = $A_{1}$ + $A_{2}$ + $A_{3}$ + $A_{4}$ + $A_{5}$ + $A_{6}$ + $A_{7}$ + $A_{8}$ + $A_{9}$ + $A_{10}$= $\frac{1}{5}$ $\times$ (0.6 + 0.8 + 0.917 + 0.98 + 1 + 0.98 + 0.917 + 0.8 + 0.6 + 0) = 1.519 Units square
