Answer
$$3e-6$$
Work Step by Step
Our aim is to integrate the triple integral as follows:
$$\int^1_0 \int^1_0 \int^1_{x^2} 12xze^{zy^2} \space dy \space dx dz \space \\= \int^1_0 \int^1_0 \int^\sqrt{y}_0 12xz e^{zy^2} \space dx \space dy \space dz \\=6 \times \int^1_0 \int^1_0 y \space z \space e^{zy^2} \space dy \space dz \\=3\times \int^1_0 [e^{zy^2}]^1_0 \space dz \\=3 \int^1_0 (e^z-1) \space dz \\\\=3e-6$$
