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Question
\[\int\limits_1^4 \left( x^2 + x \right) dx\]
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Solution
\[\text{Here }a = 1, b = 4, f\left( x \right) = x^2 + x, h = \frac{4 - 1}{n} = \frac{3}{n}\]
Therefore,
\[ \int_1^4 \left( x^2 + x \right) d x = \lim_{h \to 0} h\left[ f\left( a \right) + f\left( a + h \right) + f\left( a + 2h \right) + . . . . . . . . . . . . + f\left( a + \left( n - 1 \right)h \right) \right]\]
\[ = \lim_{h \to 0} h\left[ f\left( 1 \right) + f\left( 1 + h \right) + . . . . . . . . . . + f\left( 1 + \left( n - 1 \right)h \right) \right]\]
\[ = \lim_{h \to 0} h\left[ 1 + 1 + \left( 1 + h \right)^2 + \left( 1 + h \right) + \left( 1 + 2h \right)^2 + \left( 1 + 2h \right) + . . . . . . . . . + \left( 1 + \left( n - 1 \right)h \right)^2 + \left( 1 + \left( n - 1 \right)h \right) \right]\]
\[ = \lim_{h \to 0} h\left[ 2n + h^2 \left( 1^2 + 2^2 + . . . . . . . . . . . . . . \left( n - 1 \right)^2 \right) + 2h\left( 1 + 2 + . . . . . . + \left( n - 1 \right) \right) + h\left( 1 + 2 + . . . . . . + \left( n - 1 \right) \right) \right]\]
\[ = \lim_{h \to 0} h\left[ 2n + h^2 \frac{n\left( n - 1 \right)\left( 2n - 1 \right)}{6} + 3h\frac{n\left( n - 1 \right)}{2} \right]\]
\[ = \lim_{n \to 0 } \left[ 6 + \frac{9}{2}\left( 1 - \frac{1}{n} \right)\left( 2 - \frac{1}{n} \right) + \frac{9}{2}\left( 1 - \frac{1}{n} \right) \right]\]
\[ = 6 + 9 + \frac{9}{2} = \frac{27}{2}\]
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