<s>[INST] 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?
 [/INST] $$ \int_0^1 \int_\sqrt{y}^1 \int_0^{x^2+y^2} dz dx dy
= \int_0^1 \int_\sqrt{y}^1 ( x^2 + y^2) dx dy
$$
$$\int_\sqrt{y}^1 ( x^2 + y^2) dx=\left(\frac{1}{3}x^3+y^2x\right)_{x=\sqrt{y}}^{x=1} =\frac{1}{3}+y^2-\frac{1}{3}y^{3/2}-y^{5/2}$$
$$
\int_0^1(\frac{1}{3}+y^2-\frac{1}{3}y^{3/2}-y^{5/2})dy=\left( \frac{1}{3}y +\frac{1}{3}y^3 -\frac{2}{15}y^{5/2} -\frac{2}{7}y^{7/2}\right)_{y=0}^{y=1}=$$ $$=\frac{1}{3}+\frac{1}{3}-\frac{2}{15}-\frac{2}{7}=\frac{26}{105}$$
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