Compute the mass: \[ M = \int_{x=0}^{1} \int_{y=x^2}^{1} (x+y)\, dy\, dx. \] Reverse the order of integration: \[ M = \int_{x=0}^{1} \Big[ xy + \frac{y^2}{2} \Big]_{y=x^2}^{1} dx = \int_{0}^{1} \Big[ (x+\tfrac{1}{2}) - (x^3+\tfrac{x^4}{2}) \Big] dx. \] Integrate term by term: \[ M = \Big( \frac{x^2}{2} + \frac{x}{2} - \frac{x^4}{4} - \frac{x^5}{10} \Big)_{0}^{1} = \frac{1}{2} + \frac{1}{2} - \frac{1}{4} - \frac{1}{10}. \] Simplify: \[ M = \frac{13}{20}. \]