Answer
True.
Work Step by Step
$\int_{-1}^{1} \int_{0}^{1} e^{x^2+y^2} \sin y \, dx \, dy = \int_{0}^{1} e^{x^2} \, dx \int_{-1}^{1} e^{y^2} \sin y \, dy = \int_{0}^{1} e^{x^2} \, dx (0) = 0$. Note that $\int_{-1}^{1} e^{y^2} \sin y \, dy = 0$ because $\sin{y}$ is odd and integrating from $-1$ to $1$ will be $0$.
