ToPolarCoordinates in 3 dimensions gives hyperspherical coordinates, not cylindrical. You mention a differential equation, but I cannot see any differential equation in your calculation. You can visualize your integral this way:
reg = ImplicitRegion[0 < x < 1 &&
-Sqrt[1 - x^2] < y < Sqrt[1 - x^2] &&
Sqrt[x^2 + y^2] < z < 1,
{x, y, z}];
% // Region
Volume[reg]
Integrate[1, Element[{x, y, z}, reg]]
