Answer
For n = 2, A = 1.57 Units square
For n = 5, A = 1.934 Units square
For n = 10, A = 1.984 Units square
Work Step by Step
1. First, consider n = 2 rectangles.
This means we divide the given interval [-$\frac{\pi}{2}$, $\frac{\pi}{2}$] by 2. Thus, each subinterval has a length of $\frac{\pi}{n}$ = $\frac{\pi}{2}$.
The endpoints of the subintervals:
1) f(-$\frac{\pi}{2}$+$\frac{\pi}{2}$) = f(0) = 1
2) f($\frac{\pi}{2}$) = 0
Finally, the total area of n = 2 rectangles:
A = $A_{1}$ + $A_{2}$ = $\frac{\pi}{2}$ $\times$ (1 + 0) = 1.57 Units square
2. In the same way, we can consider n = 5 rectangles.
This means we divide the given interval [-$\frac{\pi}{2}$, $\frac{\pi}{2}$] by 5. Thus, each subinterval has a length of $\frac{\pi}{n}$ = $\frac{\pi}{5}$.
The endpoints of the subintervals:
1) f(-$\frac{\pi}{2}$+$\frac{\pi}{5}$) = f(-$\frac{3\pi}{10}$) = 0.588
2) f(-$\frac{\pi}{2}$+$\frac{2\pi}{5}$) = f(-$\frac{\pi}{10}$) = 0.951
3) f(-$\frac{\pi}{2}$+$\frac{3\pi}{5}$) = f($\frac{\pi}{10}$) = 0.951
4) f(-$\frac{\pi}{2}$+$\frac{4\pi}{5}$) = f($\frac{3\pi}{10}$) = 0.588
5) f($\frac{\pi}{2}$) = 0
Finally, the total area of n = 5 rectangles:
A = $A_{1}$ + $A_{2}$ + $A_{3}$ + $A_{4}$ + $A_{5}$ = $\frac{\pi}{5}$ $\times$ (0.588 + 0.951 + 0.951 + 0.558 + 0) = 1.934 Units square
3. In the same way, we can consider n = 10 rectangles.
This means we divide the given interval [-$\frac{\pi}{2}$, $\frac{\pi}{2}$] by 10. Thus, each subinterval has a length of $\frac{\pi}{n}$ = $\frac{\pi}{10}$.
The endpoints of the subintervals:
1) f(-$\frac{\pi}{2}$+$\frac{\pi}{10}$) = f(-$\frac{2\pi}{5}$) = 0.309
2) f(-$\frac{\pi}{2}$+$\frac{2\pi}{10}$) = f(-$\frac{3\pi}{10}$) = 0.588
3) f(-$\frac{\pi}{2}$+$\frac{3\pi}{10}$) = f(-$\frac{2\pi}{10}$) = 0.809
4) f(-$\frac{\pi}{2}$+$\frac{4\pi}{10}$) = f(-$\frac{\pi}{10}$) = 0.951
5) f(-$\frac{\pi}{2}$+$\frac{5\pi}{10}$) = f(0) = 1
6) f(-$\frac{\pi}{2}$+$\frac{6\pi}{10}$) = f($\frac{\pi}{10}$) = 0.951
7) f(-$\frac{\pi}{2}$+$\frac{7\pi}{10}$) = f($\frac{2\pi}{10}$) = 0.809
8) f(-$\frac{\pi}{2}$+$\frac{8\pi}{10}$) = f($\frac{3\pi}{10}$) = 0.588
9) f(-$\frac{\pi}{2}$+$\frac{9\pi}{10}$) = f($\frac{2\pi}{5}$) = 0.309
10) f($\frac{\pi}{2}$) = 0
Finally, the total area of n = 10 rectangles:
A = $A_{1}$ + $A_{2}$ + $A_{3}$ + $A_{4}$ + $A_{5}$ + $A_{6}$ + $A_{7}$ + $A_{8}$ + $A_{9}$ + $A_{10}$= $\frac{\pi}{10}$ $\times$ (0.309 + 0.588 + 0.809 + 0.951 + 1 + 0.951 + 0.809 + 0.588 + 0.309 + 0) = 1.984 Units square
