Evaluate the upper and lower sums for f(x) = 2 + sin(x), 0 ≤ x ≤ π, with n = 2, 4, and 8. Illustrate with diagrams like the figure shown below. (Round your answers to two decimal places.)

[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]