Answer
False, the two integrals describe different areas of integration.
Work Step by Step
Note that $\int_{0}^{1} \int_{0}^{x} \sqrt(x+y^2) \, dy \, dx \neq \int_{0}^{x} \int_{0}^{1} \sqrt(x+y^2) \, dx \, dy$ because the area of integration is different. We must switch the integration bounds to $\int_{0}^{1} \int_{y}^{1} \sqrt(x+y^2) \, dx \, dy$ to ensure the same area of integration and thus equality.
