Answer
$$\eqalign{
& \left( {\text{a}} \right){\text{Graph}} \cr
& \left( {\text{b}} \right){\text{The integrand does not have an elementary antiderivative}} \cr
& \left( {\text{c}} \right)A \approx 1.2556 \cr} $$
Work Step by Step
$$\eqalign{
& y = \sqrt x {e^x},{\text{ }}y = 0,{\text{ }}x = 0,{\text{ }}x = 1 \cr
& \cr
& \left( {\text{a}} \right){\text{Graph shown below}} \cr
& \cr
& \left( {\text{b}} \right){\text{From the area shown below we can define the area of}} \cr
& {\text{the region as:}} \cr
& A = \int_0^1 {\sqrt x {e^x}} dx \cr
& {\text{Simplifying}} \cr
& A = \int_0^1 {{x^{1/2}}{e^x}} dx \cr
& \int {\sqrt x {e^x}dx} \cr
& {\text{We can see that the area of the region is difficult to find by }} \cr
& {\text{hand because we cannot apply substitution or other method}} \cr
& {\text{the integrand does not have an elementary antiderivative}}{\text{.}} \cr
& \cr
& \left( {\text{c}} \right){\text{ From the graphing utility we can approximate the area}} \cr
& A \approx 1.2556 \cr
& \cr} $$
