Chapter 4.2, Problem 46E

Numerical Reasoning Consider a triangle of area 4 bounded by the graphs of $y=x,y=0,x=1$, and $x=3$.

(a) Sketch the region.

(b) Divide the interval [1, 3] into n subintervals of equal width and show that the endpoints are

$1<1+1\left(\frac{2}{n}\right)<\cdots <1+(n-1)\left(\frac{2}{n}\right)<1+n\left(\frac{2}{n}\right)$

(c) Show that $s(n)={\displaystyle \underset{i=1}{\overset{n}{\sum }}\left[1+(i-1)\left(\frac{2}{n}\right)\right]\left(\frac{2}{n}\right)}$

(d) show that $S(n)={\displaystyle \underset{i=1}{\overset{n}{\sum }}\left[1+i\left(\frac{2}{n}\right)\right]\left(\frac{2}{n}\right)}$

(e) Find s(n) and S(n) for $n=5,10,50$, and 100.

(f) Show that $\underset{n\to \infty }{\lim }s(n)=\underset{n\to \infty }{\lim }S(n)=4$