Evaluate $\int_0^1 \int_\sqrt{y}^1 \int_0^{x^2+y^2} dz dx dy$. 
Evaluate $\int_0^1 \int_\sqrt{y}^1 \int_0^{x^2+y^2} dz dx dy$.

Attempt:
$$ \int_0^1 \int_\sqrt{y}^1 \int_0^{x^2+y^2} dz dx dy
= \int_0^1 \int_\sqrt{y}^1  x^2 + y^2 dx dy
= 1/3 + 1/3 - 2/15 - 2/7 = \frac{26}{105}.$$
However, the solution should be $26/35$ so it appears that I'm off by a factor of $3$ but I don't see where this fits in. Where did I go wrong?

There is nothing wrong with your calculations.
$$\int_0^1 \int_{y^{1/2}}^1 x^2 + y^2 \ dx \ dy = \int_0^1 y^2 - y^{5/2} + \frac13(1-y^{3/2}) \ dy \\
 = \frac 13 - \frac 27 + \frac 13 - \frac 2{15} = \frac{26}{105}$$
