Triple integral evaluation, I'm having difficulty understanding the change of the order of integration.

Question

$$\int_{0}^1\int_{0}^z\int_{z}^1e^{y^3}dydxdz + \int_{0}^1\int_{z}^1\int_{x}^1e^{y^3}dydxdz$$ is the problem. The solution states that the result of the sum of the two is $$\int_{0}^1\int_{0}^y\int_{0}^ye^{y^3}dxdzdy$$ which I can't wrap my head around. Exactly how is the 'conversion' from dydxdz to dxdzdy achieved?

Answer 1

For each fixed $z\in [0,1]$ we have $$\int_0^z \int _z^1 e^{y^3}\mathrm{d}y\mathrm{d}x+\int_z^1 \int_x^1 e^{y^3}\mathrm{d}y\mathrm{d}x=\int_z^1\int_0^ye^{y^3}\mathrm{d}x\mathrm{d}y$$ Hence $$\begin{eqnarray*}\int_0^1 \int_0^z \int _z^1 e^{y^3}\mathrm{d}y\mathrm{d}x\mathrm{d}z+\int_0^1\int_z^1 \int_x^1 e^{y^3}\mathrm{d}y\mathrm{d}x\mathrm{d}z &=& \int_0^1 \Bigg[\int_0^z \int _z^1 e^{y^3}\mathrm{d}y\mathrm{d}x+\int_z^1 \int_x^1 e^{y^3}\mathrm{d}y\mathrm{d}x\Bigg]\mathrm{d}z \\ &=& \int_0^1 \int_z^1 \int_0^y e^{y^3}\mathrm{d}x\mathrm{d}y\mathrm{d}z \\ &=& \int_0^1 \int _0^y \int_0^y e^{y^3}\mathrm{d}x\mathrm{d}z\mathrm{d}y\end{eqnarray*}$$