Evaluating $\int_0^1\int_0^1 e^{\max\{x^2,y^2\}\,}\mathrm dx\,\mathrm dy$

Do not use polar coordinates: Rectangles are bad for polar coordinates.

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You have a piecewise definition that's making things difficult, so use each part of the domain separately. Break the integral up into two regions, above and below the diagonal of the unit square. On the lower half,

$$\int_0^1 \int_0^x e^{\max\{x^2, y^2\}} dy \, dx = \int_0^1 \int_0^x e^{x^2} \, dy \, dx = \int_0^1 xe^{x^2} \, dx$$

The upper half is similar.

HINT:

Note that

$$I=\int_0^1\int_0^y e^{y^2}\,dx\,dy+\int_0^1\int_y^1 e^{x^2}\,dx\,dy$$

Now, interchange the order of integration in the second integral.