Integration in cylindrical coordinates

 Objective: Evaluate a triple integral by changing to cylindrical coordinates

Review of Cylindrical coordinates

Representation of a point in polar coordinates in a two-dimensional space

A point (x,y) in rectangular coordinates is represented in polar coordinates by (r, θ) and vice-versa. The relationships between the variables are represented as follow:

Representation of a point in cylindrical coordinates in a three-dimensional space

A point (x, y, z) in rectangular coordinates is represented in cylindrical coordinates by (r, θ, z) and vice-versa. The same relations above are represented with z representing the vertical distance of the point P(x,y,z) to the xy plane as shown in the following figure:

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 Conversion from rectangular coordinates to cylindrical coordinates

 In a two-dimensional space in rectangular coordinates the line x = l represents a line parallel to the y axis and the line y = k represents a line parallel to the x axis. In polar coordinates, r =c means a circle of radius c units. The expression θ = ɑ means the angle made by an infinite ray with the positive direction of the x axis.

In a three-dimensional space, when we consider a point with cylindrical coordinates (r, θ, z)., r =c means a horizontal plane parallel to the xz plane, θ represents a plane making a constant angle with the xy plane and z = m the radial axis of a cylinder.

Integration in cylindrical coordinates

Let's consider a simple bounded region B in R³ in the form of a cylindrical box.

Suppose we divide each interval into l, m and n subdivisions such that 

This allows to state following definition of integrals in cylindrical coordinates based on the following figure:

 

Definition

Let's consider the cylindrical box expressed by:

If g(x,.y, z) is the function expressed in rectangular coordinates and the box B is in rectangular coordinates, then

Theorem

Example

Solution

Eliminating the parenthesis and integrating each variable separately, we obtain:

Practice