The position of a point M (x, y, z) in the xyz-space in cylindrical coordinates is defined by three numbers: ρ, φ, z, where ρ is the projection of the radius vector of the point M onto the xy-plane, φ is the angle formed by the projection of the radius vector with the x-axis (Figure 1), z is the projection of the radius vector on the z-axis (its value is the same in Cartesian and cylindrical coordinates).

Figure 1.

The relationship between cylindrical and Cartesian coordinates of a point is given by

Transition from cylindrical coordinates makes calculation of triple integrals simpler in those cases when the region of integration is formed by a cylindrical surface.

Solved Problems

Example 1.

Evaluate the integral \[\iiint\limits_U {\left( {{x^4} + 2{x^2}{y^2} + {y^4}} \right)dxdydz},\] where the region \(U\) is bounded by the surface \({x^2} + {y^2} \le 1\) and the planes \(z = 0,\) \(z = 1\) (Figure \(2\text{).}\)

Solution.

Figure 2.Figure 3.

It is more convenient to calculate this integral in cylindrical coordinates. Projection of the region of integration onto the \(xy\)-plane is the circle \({x^2} + {y^2} \le 1\) or \(0 \le \rho \le 1\) (Figure \(3\)).

The second integral contains the factor \(\rho\) which is the Jacobian of transformation of the Cartesian coordinates into cylindrical coordinates. All the three integrals over each of the variables do not depend on each other. As a result the triple integral is easy to calculate as

The projection of the region of integration \(U\) onto the \(xy\)-plane is the circle \({x^2} + {y^2} \le 9\) with radius \(\rho = 3\) (Figure \(5\)). The coordinate \(\rho\) ranges from \(0\) to \(3,\) the angle \(\varphi\) ranges from \(0\) to \(2\pi,\) and the coordinate \(z\) ranges from \(\frac{{{\rho ^2}}}{3}\) to \(3.\)