Answer
$\int_0^1 \int_{0.5 y^2}^{y^2} y e^{1+x} \ dx \ dy=0.5 (e^2-2e^{1.5}+e)$
Work Step by Step
Here, we have: $\int_0^1 \int_{0.5 y^2}^{y^2} y e^{1+x} \ dx \ dy=\int_0^1 [y e^{1+x}]_{0.5 y^2}^{y^2} \ dy$
or, $= [\dfrac{1}{2} \times e^{1+y^2}-e^{1+0.5y^2}]_0^1 $
or, $=\dfrac{e^2}{2}-e^{1.5}-\dfrac{e}{2}+e$
or, $=\dfrac{e^2}{2}-e^{1.5}+\dfrac{e}{2}$
or, $=0.5 (e^2-2e^{1.5}+e)$
Thus, we get:
$\int_0^1 \int_{0.5 y^2}^{y^2} y e^{1+x} \ dx \ dy=0.5 (e^2-2e^{1.5}+e)$
