Answer
$A \approx 3.6601$
Work Step by Step
$$\eqalign{
& y = {e^{1 - {x^2}}},{\text{ }}y = {x^4} \cr
& {\text{Graph the curves using Geogebra }}\left( {{\text{Shown Below}}} \right) \cr
& {\text{Find the intersection points using the graph we obtain}} \cr
& x = - 1,{\text{ }}x = 1 \cr
& {\text{We have the interval }}\left[ { - 1,1} \right] \cr
& {\text{We can find the area integrating with respect to }}x \cr
& A = \int_a^b {\left[ {f\left( x \right) - g\left( x \right)} \right]} dx{\text{ }}\left( {\bf{1}} \right){\text{ }}\left( {{\text{see page 439}}} \right) \cr
& {\text{From the graph }} \cr
& {e^{1 - {x^2}}} \geqslant {x^4}{\text{ for the interval }}\left[ { - 1,1} \right] \cr
& {\text{Therefore}} \cr
& A = \int_{ - 1}^1 {\left( {{e^{1 - {x^2}}} - {x^4}} \right)} dx \cr
& or \cr
& A = 2\int_0^1 {\left( {{e^{1 - {x^2}}} - {x^4}} \right)} dx \cr
& {\text{Integrating by a graphing calculator we obtain}} \cr
& A \approx 3.6601 \cr} $$
