Evaluate ${\int }_1^3 \left( 3 {\mathrm{x}}^2 + 2 \mathrm{x} \right) \mathrm{dx}$  as limit of sums.

$\mathrm{I} = {\int }_1^3 \left( 3 {\mathrm{x}}^2 + 2\mathrm{x} \right) \mathrm{dx}$

Here    a = 1,   b = 3

$$\begin{gathered} \mathrm{f} ( \mathrm{x} ) = 3 {\mathrm{x}}^2 + 2\mathrm{x} \mathrm{h} = \frac{\mathrm{b} - \mathrm{a}}{\mathrm{n}} = \frac{2}{\mathrm{n}} \\ \mathrm{Since}, {\int }_{\mathrm{a}}^{\mathrm{b}} \mathrm{f} ( \mathrm{x} ) \mathrm{dx} = \underset{\mathrm{h} \to 0}{\lim } \mathrm{h} \left[ \mathrm{f} ( \mathrm{a} ) + \mathrm{F} ( \mathrm{a} + \mathrm{h} ) ......+ \mathrm{f} ( + ( \mathrm{n} - 1 ) \mathrm{h} )\right] \\ \mathrm{so}, {\int }_1^3 \left( 3 {\mathrm{x}}^2 + 2\mathrm{x} \right) = \underset{\mathrm{h} \to 0}{\lim } \mathrm{h} [ ( 3 ( 1 {)}^2 + 2 ( 1 ) ) + ( 3 ( 1 +\mathrm{h} {)}^2 + 2 ( 1 +\mathrm{h} ) ) + \\ 3 ( 1 + 2\mathrm{h} {)}^2 + 2 ( 1 + 2\mathrm{h} ) )..........+ 3 ( 1 + ( \mathrm{n} - 1 ) \mathrm{h} {)}^2 + 2 ( 1 + ( \mathrm{n} - 1 ) \mathrm{h} ) ] \\ = \underset{\mathrm{h} \to 0}{\lim } \mathrm{h} [ 3 ( \mathrm{n} ) + 3 ( {\mathrm{h}}^2 + 4 {\mathrm{h}}^2 + .........+ ( \mathrm{n} - 1 {)}^2 {\mathrm{h}}^2 ) + 3 ( 2\mathrm{h} + 4\mathrm{h} + .......+ \\ 2 ( \mathrm{n} - 1 ) \mathrm{h} ) + 2 \mathrm{n} + 2 ( \mathrm{h} + 2 \mathrm{h} + ..........+ ( \mathrm{n} - 1 ) \mathrm{h} ) ] \\ = \underset{\mathrm{h} \to 0}{\lim } \mathrm{h} [ 5 \mathrm{n} \mathrm{h} + 3 {\mathrm{h}}^3 ( 1^2 + 2^2 + .....+ ( \mathrm{n} - 1 {)}^2 + 6 {\mathrm{h}}^2 ( 1 + 2 + ...........+ ( \mathrm{n} - 1 ) )+ \\ 2 {\mathrm{h}}^2 ( 1 + 2 + ( \mathrm{n} - 1 ) ) ] \end{gathered}$$

           $$\begin{gathered} = \underset{\mathrm{h} \to 0}{\lim } \left[ 5 \mathrm{n} \mathrm{h} + 3{\mathrm{h}}^3 \times \frac{( \mathrm{n} - 1 ) \mathrm{n} ( 2\mathrm{n} - 1 )}{6} + \frac{8 {\mathrm{h}}^2 ( \mathrm{n} ) ( \mathrm{n} - 1 )}{2}\right] \\ = \underset{\mathrm{h} \to 0}{\lim } \left[ 10 + \frac{ \left(\mathrm{n} \mathrm{h} - \mathrm{h} \right) \mathrm{n} \mathrm{h} \left( 2 \mathrm{n} \mathrm{h} - \mathrm{h} \right)}{2} + 4 ( \mathrm{n} \mathrm{h} ) ( \mathrm{n} \mathrm{h} - \mathrm{h} ) \right] \\ = \left[ 10 + \frac{2 \times \cancel{2} \times 4}{\cancel{2}} + 4 \mathrm{x} 2 \mathrm{x} 2 \right] \\ = 10 + 8 + 16 = 34 \end{gathered}$$