Chapter 4 - Integration - 4.3 Exercises - Page 276: 71

$$\frac{{{b^2} - {a^2}}}{2}$$

$$\eqalign{ & \int_a^b x dx = \frac{{{b^2} - {a^2}}}{2} \cr & {\text{Using the definition of Definite Integrals }}\left( {{\text{Page 268}}} \right) \cr & \Delta x = \frac{{b - a}}{n}{\text{ and }}{c_i} = a + i\left( {\Delta x} \right),{\text{ }}{c_i} = a + i\left( {\frac{{b - a}}{n}} \right) \cr & \int_a^b {f\left( x \right)} dx = \mathop {\lim }\limits_{\left\| \Delta \right\| \to 0} \sum\limits_{i = 1}^n {f\left( {{c_i}} \right)} \Delta {x_i} \cr & {\text{Therefore, }} \cr & \int_a^b x dx = \mathop {\lim }\limits_{\left\| \Delta \right\| \to 0} \sum\limits_{i = 1}^n {f\left( {{c_i}} \right)} \Delta {x_i} \cr & {\text{ }} = \mathop {\lim }\limits_{n \to \infty } \sum\limits_{i = 1}^n {\left[ {a + i\left( {\frac{{b - a}}{n}} \right)} \right]} \left( {\frac{{b - a}}{n}} \right) \cr & {\text{ }} = \mathop {\lim }\limits_{n \to \infty } \sum\limits_{i = 1}^n {\left[ {a\left( {\frac{{b - a}}{n}} \right) + i{{\left( {\frac{{b - a}}{n}} \right)}^2}} \right]} \cr & {\text{Properties of a summation}} \cr & {\text{ }} = \mathop {\lim }\limits_{n \to \infty } \left[ {\sum\limits_{i = 1}^n {a\left( {\frac{{b - a}}{n}} \right)} + \sum\limits_{i = 1}^n {{{\left( {\frac{{b - a}}{n}} \right)}^2}} i} \right] \cr & {\text{ }} = \mathop {\lim }\limits_{n \to \infty } \left[ {\left( {\frac{{b - a}}{n}} \right)\sum\limits_{i = 1}^n a + {{\left( {\frac{{b - a}}{n}} \right)}^2}\sum\limits_{i = 1}^n i } \right] \cr & {\text{Using the summation Formulas THEOREM 4}}{\text{.2}} \cr & {\text{ }} = \mathop {\lim }\limits_{n \to \infty } \left[ {a\left( {\frac{{b - a}}{n}} \right)n + {{\left( {\frac{{b - a}}{n}} \right)}^2}\left( {\frac{{n\left( {n + 1} \right)}}{2}} \right)} \right] \cr & {\text{ }} = \mathop {\lim }\limits_{n \to \infty } \left[ {a\left( {b - a} \right) + {{\left( {\frac{{b - a}}{n}} \right)}^2}\left( {\frac{{n + 1}}{2}} \right)} \right] \cr & {\text{ }} = \mathop {\lim }\limits_{n \to \infty } \left[ {a\left( {b - a} \right) + \frac{{{{\left( {b - a} \right)}^2}}}{n}\left( {\frac{{n + 1}}{2}} \right)} \right] \cr & {\text{ }} = \mathop {\lim }\limits_{n \to \infty } \left[ {a\left( {b - a} \right) + \frac{{{{\left( {b - a} \right)}^2}}}{n}\left( {\frac{1}{2} + \frac{1}{{2n}}} \right)} \right] \cr & {\text{Evaluate the limit when }}n \to \infty \cr & {\text{ }} = a\left( {b - a} \right) + \frac{{{{\left( {b - a} \right)}^2}}}{n}\left( {\frac{1}{2} + \frac{1}{\infty }} \right) \cr & {\text{ }} = a\left( {b - a} \right) + \frac{{{{\left( {b - a} \right)}^2}}}{2} \cr & {\text{Factoring}} \cr & {\text{ }} = \left( {b - a} \right)\left( {a + \frac{{b - a}}{2}} \right) \cr & {\text{ }} = \left( {b - a} \right)\left( {\frac{{a + b}}{2}} \right) \cr & {\text{ }} = \frac{{{b^2} - {a^2}}}{2} \cr} $$