# Triple Integrals in Cylindrical Coordinates

Calculus 3: Triple Integrals in Cylindrical Coordinates (Video #24)
Calculus 3: Triple Integrals in Cylindrical Coordinates (Video #24)

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 $$3.$$

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