Answer
$$\eqalign{
& \left( {\text{a}} \right){\text{Graph}} \cr
& \left( {\text{b}} \right){\text{The intersection points are difficult to find by hand}}. \cr
& \left( {\text{c}} \right)A \approx 3.0577 \cr} $$
Work Step by Step
$$\eqalign{
& y = {x^2},{\text{ }}y = \sqrt {3 + x} \cr
& \cr
& \left( {\text{a}} \right){\text{Graph shown below}} \cr
& \cr
& \left( {\text{b}} \right){\text{From the area shown below we can define the area of}} \cr
& {\text{the region as:}} \cr
& A = \int_{ - a}^a {\left( {\sqrt {3 + x} - {x^2}} \right)} dx \cr
& {\text{Find the intersection points }}a{\text{ and }}b \cr
& \sqrt {3 + x} = {x^2} \cr
& 3 + x = {x^4} \cr
& {x^4} - x - 3 = 0{\text{ }}\left( {{\text{Hard to find by hand}}} \right) \cr
& {\text{We can see that the antiderivative of the integrand}}{\text{, is easy}} \cr
& {\text{to calculate using the elementary antiderivative}}{\text{, but the}} \cr
& {\text{intersection points }}x,{\text{ it is hard because the equation}} \cr
& {x^4} - x - 3 = 0{\text{ it is difficult to find by hand}}. \cr
& \cr
& \left( {\text{c}} \right){\text{ From the graphing utility we can approximate the area}} \cr
& A \approx 3.0577{\text{ and with }}a = - 1.1640{\text{ and }}b = 1.4526 \cr} $$
