Answer
$$A = \frac{3}{{10}}$$
Work Step by Step
$$\eqalign{
& {\text{Let }}f\left( x \right) = x,{\text{ }}g\left( x \right) = - {x^4} \cr
& {\text{From the graph shown below}} \cr
& g\left( x \right) \geqslant f\left( x \right){\text{ on the interval }} - 1 \leqslant x \leqslant 0 \cr
& {\text{The area is given by}} \cr
& A = \int_{ - 1}^0 {\left( { - {x^4} - x} \right)} dx \cr
& {\text{ Integrate}} \cr
& A = \left[ { - \frac{1}{5}{x^5} - \frac{1}{2}{x^2}} \right]_{ - 1}^0 \cr
& A = \left[ { - \frac{1}{5}{{\left( 0 \right)}^5} - \frac{1}{2}{{\left( 0 \right)}^2}} \right] - \left[ { - \frac{1}{5}{{\left( { - 1} \right)}^5} - \frac{1}{2}{{\left( { - 1} \right)}^2}} \right] \cr
& {\text{Simplifying}} \cr
& A = \left( 0 \right) - \left( { - \frac{3}{{10}}} \right) \cr
& A = \frac{3}{{10}} \cr} $$
