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