Chapter 5 - The Integral - 5.1 Approximating and Computing Area - Exercises - Page 236: 47

$$18 $$

Given $$f(x)=9 x, \quad[0,2]$$ Since $\Delta x= \dfrac{b-a}{N}=\dfrac{2}{N}$, and $$ x_i =a+i\Delta x= \dfrac{2i}{N}$$ Since \begin{align*} R_N& =\Delta x \sum_{i=1}^{N} f\left(x_{i}\right)\\ &=\frac{2}{N} \sum_{i=1}^{N} 9\left(\frac{2 i}{N}\right)\\ &=\frac{36}{N^{2}} \sum_{i=1}^{N} i\\ &=\frac{36}{N^{2}}\left(\frac{N(N+1)}{2} \right) \end{align*} Then \begin{align*} \lim_{N\to\infty}R_N&=\lim_{N\to\infty}\frac{36}{N^{2}}\left(\frac{N(N+1)}{2} \right)\\ &=\lim_{N\to\infty} \left(\frac{18N^2+18}{N^2} \right)\\ &=18 \end{align*}