Question

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.)

Answer 1

$f(x)=x+sin(x)\\ a=0\\ b=\pi \\ x_i=a+idelta(x)\\ Upper sum for n=2:\\ \\ delta(x)=\frac{b-a}{n} =\frac{\pi-0}{2} =\frac{\pi}{2}$

$x_0=0, x_1=\frac{\pi}{2} ,x_1=\pi$

Length of the subintervals: $[0,\frac{\pi}{2}], [\frac{\pi}{2}, \pi}]$

Using Upper Riemann sum,

$\int\limits^0_\pi{x+sin(x)\, dx =$∑Max{f(x_i) delta(x)

$=[max{(f(x_1))+max(f(x_2))]delta(x)\\ =(3+3)\frac{\pi}{2} \\=9.42$

Lower sum for n=2:

The minimum value for the function on $[0,\frac{\pi}{2}], [\frac{\pi}{2}, \pi]$ is 2.

$\int\limits^0_\pi {x+\sin x} \, dx =\sum_{n=0}^{n=2} min {f(x_i)} delta (x)$

$= (2+2)\frac{\pi}{2}\\ =6.28$