Official solution
Equation of tangent at (2, 3) on the parabola y = x2 - 1 is
\[{{y + 3} \over 2} = 2x - 1\]
\[ \Rightarrow \] y + 3 = 4x - 2
\[ \Rightarrow \] y = 4x - 5
When x = 0 then for the tangent y = - 5
\[ \therefore \] Tangent cuts x y axis at (0, - 5) point.
\[ \therefore \] Area of the bounded region is
= \[\int\limits_{ - 5}^3 {{{y + 5} \over 4}} \,\,\,dy - \int\limits_{ - 1}^3 {\sqrt {y + 1} } \,\,\,dy\]
= \[{1 \over 4}\left[ {{{{y^2}} \over 2} + 5y} \right]_{ - 5}^3 - \left[ {{2 \over 3} \times {{\left( {y + 1} \right)}^{{3 \over 2}}}} \right]_{ - 1}^3\]
\[{1 \over 4}\left[ {\left( {{9 \over 2} + 15} \right) - \left( {{{25} \over 2} - 25} \right)} \right] - {2 \over 3}{\left( 4 \right)^{{3 \over 2}}}\]
= \[{1 \over 4}\left[ {{{93} \over 2} + {{25} \over 2}} \right] - {2 \over 3} \times 8\]
= \[{1 \over 4} \times {{64} \over 2} - {{16} \over 3}\]
= \[8 - {{16} \over 3}\]
= \[{8 \over 3}\]