Official solution
A = \[\int\limits_{ - 1}^1 {\left( {\left( {1 - {x^2}} \right) - \left( {{x^2} - 1} \right)} \right)dx} \]
= \[\int\limits_{ - 1}^1 {\left( {2 - 2{x^2}} \right)dx} \]
= \[4\int\limits_0^1 {\left( {1 - {x^2}} \right)dx} \]
= \[4\left( {x - {{{x^3}} \over 3}} \right)_0^1\]
= \[4\left( {{2 \over 3}} \right)\] = \[{8 \over 3}\]