[tex] f(x)=x+sin(x)\\ a=0\\ b=\pi \\ x_i=a+idelta(x)\\ Upper sum for n=2:\\ \\ delta(x)=\frac{b-a}{n} =\frac{\pi-0}{2} =\frac{\pi}{2} [/tex]
[tex] x_0=0, x_1=\frac{\pi}{2} ,x_1=\pi \\ [/tex]
Length of the subintervals: [tex] [0,\frac{\pi}{2}], [\frac{\pi}{2}, \pi}] [/tex]
Using Upper Riemann sum,
[tex] \int\limits^0_\pi{x+sin(x)\, dx = [/tex]∑Max{f(x_i) delta(x)
[tex] =[max{(f(x_1))+max(f(x_2))]delta(x)\\ =(3+3)\frac{\pi}{2} \\=9.42 [/tex]
Lower sum for n=2:
The minimum value for the function on [tex] [0,\frac{\pi}{2}], [\frac{\pi}{2}, \pi] [/tex] is 2.
[tex] \int\limits^0_\pi {x+\sin x} \, dx =\sum_{n=0}^{n=2} min {f(x_i)} delta (x)\\
[/tex]
[tex] = (2+2)\frac{\pi}{2}\\ =6.28 [/tex]