Consider a triangle of area 2 bounded by the graphs of and (a) Sketch the region. (b)

Chapter 4, Problem 55

(choose chapter or problem)

Numerical Reasoning Consider a triangle of area 2 bounded by the graphs of y=x, y=0,and x=2

(a) Sketch the region.

(b) Divide the interval [0,2] into n subintervals of equal width and show that the endpoints are

\(0<1\left(\frac{2}{n}\right)<\cdots<(n-1)\left(\frac{2}{n}\right)<n\left(\frac{2}{n}\right)\)

(c) Show that \(s(n)=\sum_{i=1}^{n}\left[(i-1)\left(\frac{2}{n}\right)\right]\left(\frac{2}{n}\right) .\)

(d) Show that \(S(n)=\sum_{i=1}^{n}\left[i\left(\frac{2}{n}\right)\right]\left(\frac{2}{n}\right)\)

(e) Complete the table.

(f) Show that \(\lim _{n \rightarrow \infty} s(n)=\lim _{n \rightarrow \infty} S(n)=2\)

Text Transcription:

0<1(\frac{2}{n})<\cdots<(n-1)(\frac{2}{n})<n(\frac{2}{n})

s(n)=\sum_{i=1}^{n}[(i-1)(\frac{2}{n})](\frac{2}{n}) .

S(n)=\sum_{i=1}^{n}[i(\frac{2}{n})](\frac{2}{n})

\lim _{n \rightarrow \infty} s(n)=\lim _{n \rightarrow \infty} S(n)=2