Solution:

$\begin{aligned} \text { let } \mathrm{I} &=\int_{0}^{1} \int_{0}^{x^{2}} e^{\frac{y}{x}} d y d x \\ &=\int_{0}^{1}\left[\frac{e^{y}}{\frac{1}{x}}\right]_{0}^{x^{2}} d x \end{aligned}$

$=\int_{0}^{1} \frac{\left(e^{x}-1\right)}{\frac{1}{x}} d x$

$=\int_{0}^{1} x \cdot e^{x} d x-\int_{0}^{1} x \cdot d x$

$=\left[x \cdot e^{x}-e^{x}\right]_{0}^{1}-\left[\frac{x^{2}}{2}\right]_{0}^{1}$

$=e-e+1-\frac{1}{2}$

$I=\frac{1}{2}$