Question

\[\int\limits_0^3 \left( x^2 + 1 \right) dx\]

Answer 1

\[\text{Here, }a = 0, b = 3, f\left( x \right) = x^2 + 1, h = \frac{3 - 0}{n} = \frac{3}{n}\]
Therefore,
\[ \int_0^3 \left( x^2 + 1 \right) d x = \lim_{h \to 0} h\left[ f\left( a \right) + f\left( a + h \right) + f\left( a + 2h \right) + . . . . . . . . . . . . + f\left( a + \left( n - 1 \right)h \right) \right]\]
\[ = \lim_{h \to 0} h\left[ f\left( 0 \right) + f\left( 0 + h \right) + . . . . . . . . . . + f\left( 0 + \left( n - 1 \right)h \right) \right]\]
\[ = \lim_{h \to 0} h\left[ 0 + 1 + h^2 + 1 + \left( 2h \right)^2 + 1 + . . . . . . . . . + \left( \left( n - 1 \right)h \right)^2 + 1 \right]\]
\[ = \lim_{h \to 0} h\left[ n + h^2 \left( 1^2 + 2^2 + . . . . . . . . . . . . . . \left( n - 1 \right)^2 \right) \right]\]
\[ = \lim_{h \to 0} h\left[ n + h^2 \frac{n\left( n - 1 \right)\left( 2n - 1 \right)}{6} \right]\]
\[ = \lim_{h \to 0} \left[ 3 + \frac{9}{2}\left( 1 - \frac{1}{n} \right)\left( 2 - \frac{1}{n} \right) \right]\]
\[ = 3 + 9 = 12\]