Answer
$ \approx 0.0512$
Work Step by Step
Consider $I=\iint_{D} y^2 dA$
$\implies =\int_{-0.7146}^{0} \int_{e^x} ^{1-x^2} y^2 dy dx$
$\implies =\int_{-0.7146}^{0} [\dfrac{y^3}{3}]_{e^x} ^{1-x^2} dx$
$\implies=\dfrac{1}{3} \int_{-0.7146}^{0} (1-x^6 +3x^4-3x^2)-e^{3x} dx$
$\implies= [x-\dfrac{x^7}{7}+\dfrac{3x^5}{5}-x^3-\dfrac{e^{3x}}{3}]_{-0.7146}^{0} \times \dfrac{1}{3} $
Now, we will use calculator, so we have
$I \approx 0.0512$
