Because there is no θ in ρ integral, ρ integral can be regarded as a constant in θ integral, so it can be multiplied. If the range of integration contains θ, such as 0 to θ, or there is θ in the function, you can't multiply in turn and must integrate.

meaning

When the integrand function is greater than zero, the double integral is the volume of the cylinder.

When the integrand function is less than zero, the double integral is the negative value of the cylinder volume. ?

In the spatial rectangular coordinate system, the double integral is the algebraic sum of the cylindrical volumes of each part of the region, which is positive above the xoy plane and negative below the xoy plane. The volume formula of the curved cylinder surrounded by the curved surface of some special integrand f(x, y) and the bottom surface of d is known, which can be calculated by the geometric meaning of double integral.