Chapter 4 - Integration - 4.2 Exercises - Page 264: 47

$${\text{Area}} = \frac{7}{3}$$

$$\eqalign{ & y = {x^2} + 2,{\text{ }}\left[ {0,1} \right] \cr & f\left( x \right) = {x^2} + 2 \cr & f\left( 0 \right) = 2{\text{ and }}f\left( 1 \right) = 3,{\text{ }} \cr & f\left( x \right){\text{ is continuous and there are no negatives on the interval }} \\ & {\text{Area}} = \mathop {\lim }\limits_{n \to \infty } \sum\limits_{i = 1}^n {f\left( {{c_i}} \right)} \Delta x,{\text{ }}\Delta x = \frac{{1 - 0}}{n} = \frac{1}{n} \cr & {\text{Area}} = \mathop {\lim }\limits_{n \to \infty } \sum\limits_{i = 1}^n {\left[ {{{\left( {\frac{i}{n}} \right)}^2} + 2} \right]} \left( {\frac{1}{n}} \right) \cr & {\text{Area}} = \mathop {\lim }\limits_{n \to \infty } \sum\limits_{i = 1}^n {\left( {\frac{{{i^2}}}{{{n^3}}}} \right)} + \mathop {\lim }\limits_{n \to \infty } \frac{2}{n}\sum\limits_{i = 1}^n {\left( 1 \right)} \cr & {\text{Area}} = \mathop {\lim }\limits_{n \to \infty } \frac{1}{{{n^3}}}\sum\limits_{i = 1}^n {\left( {{i^2}} \right)} + 2 \cr & {\text{Where }}\sum\limits_{i = 1}^n {{i^2}} = \frac{{n\left( {n + 1} \right)\left( {2n + 1} \right)}}{6} \cr & {\text{Area}} = \mathop {\lim }\limits_{n \to \infty } \frac{1}{{{n^3}}}\left( {\frac{{n\left( {n + 1} \right)\left( {2n + 1} \right)}}{6}} \right) + 2 \cr & {\text{Area}} = \mathop {\lim }\limits_{n \to \infty } \left( {\frac{{\left( {n + 1} \right)\left( {2n + 1} \right)}}{{6{n^2}}}} \right) + 2 \cr & {\text{Area}} = \mathop {\lim }\limits_{n \to \infty } \left( {\frac{{2{n^2} + n + 2n + 1}}{{6{n^2}}}} \right) + 2 \cr & {\text{Area}} = \mathop {\lim }\limits_{n \to \infty } \left( {\frac{1}{3} + \frac{1}{{2n}} + \frac{1}{{6{n^2}}}} \right) + 2 \cr & {\text{Evaluate the limit when }}n \to \infty \cr & {\text{Area}} = \frac{1}{3} + 0 + 0 + 2 \cr & {\text{Area}} = \frac{7}{3} \cr} $$