Triple integrals from rectangular to spherical. How do you get phi? φ

You're working with a region bounded by a sphere of radius √8 and the cone z = √(x²+y²). The cone gives you the bounds on φ, the angle from the positive z-axis. At each fixed z the height of the cone is equal to its radius, forming 45-45-90 triangles. So 0 < φ < π/4.