Evaluating a triple integral in spherical coordinates

 Integration in spherical coordinates

Let f (⍴, θ, ψ) be a function continuous over a bounded spherical box defined by:

Let's divide each interval in l, m. n subintervals such that 

Let's consider any sample point (ρᵢⱼₖ, θᵢⱼₖ, ψᵢⱼₖ) in the subbox Bᵢⱼₖ. The volume element ΔV of the subbox B can be written in spherical coordinates by:

as shown in the following figure:

Let's take the Rieman sum of the expression:

The limit of this expression when l, m, n approach infinity is the triple integral of the function in spherical coordinates as defined above

Definition of a triple integral in spherical coordinates

The properties already examined for previous integrals work for triple integrals in cylindrical coordinates as well as iterated integrals. As always, Fubini's theorem allows us to evaluate a triple a integral by setting it up as an iterated integral. The theorem is stated below:

Theorem

Example

Solution

The variables being independent of each other, we can integrate each piece and multiply:

Integration in spherical coordinates

After spending some time setting up and evaluating triple integrals in cylindrical coordinates, we are going to work with triple integrals in spherical coordinates. Before doing this, let's get an overview of spherical coordinates 

Overview of spherical coordinates 

In a three-dimensional system of coordinates, a point P (x, y, z) is defined by:

ρ : the distance from the origin to the point P

θ: the angle from the positive direction of the x-axis as in the cylindrical coordinates system

ѱ: the angle from the positive z axis and the line OP.

Relationships between rectangular coordinates and spherical coordinates

Other important relationships for conversion

The following figures show a few regions that are useful to express in spherical coordinates

Finding the volume with triple integrals in three ways

 In a previous example I showed how to set up a triple integral in three ways. Calculating a volume in three ways comes to using the same procedure. The following example shows how to calculate the volume with triple integrals in three ways.

Example

Let E be the region bounded below by the rθ plane, above by the sphere x² + y² + z² = 4 and on the sides by the cylinder x² + y² = .1. Set up a triple integral in cylindrical coordinates to find the volume of the region using the following orders of integration, and in each case find the volume and check that the answers are the same.

a. dzdrdθ

b.drdzdθ

Solution

In the expression of E₁ below, r is a function of z. Therefore, we have 0 ≤ r ≤ ⎷4-z² and not 0 ≤ r ≤ ⎷4-r²

Practice

Redo the previous example with the following order dθdzdr,

Setting up a triple integral in cylindrical coordinates in two ways.

 It is possible to set up a triple integral in cylindrical coordinates in two ways. We are going to do that by solving an example.

Example 

Let E be the region bounded below by the the cone z = ⎷x² + y² and the paraboloid z = 2 - x³ - y². Set up a triple integral in cylindrical coordinates to find the volume of the region, using the following orders of integration.

a.dzdrdθ

b. drdzdθ

Solution

a. Integration following the order dzdrdθ

Let's find the intersection of the 2 surfaces. by equalizing their equation

 2 - x2 - y2 = √(x2 + y2)

We want to solve the equation:

2 − x² − y² = √(x² + y²)

2 −( x² +y²⁾ = √(x² + y²)

Since x² + y² = r²

r = √ (x² + y²)

Let's substitute x² + y² and √ (x² + y²) in the equation:

2 − r2 = r 

2 − r² − r = 0

r² + r − 2 = 0

:(r + 2) (r − 1) = 0

 r = −2 (not valid since r ≥ 0)

r = 1 (valid)

Substitute r in the expression: x² + y² = r²

 We finally have: x² + y² = 1

The equation represents a circle centered at the origin and of radius 1 The projection of the region E onto the xy plane is a circle centered at the origin and of radius 1.

Let's find the limits. For fixed θ and r, we have:

Limits for θ: 0 ≤ θ ≤ 2ℼ

Limits for r: 0 ≤ r ≤1

Limits for z:

The cone is the lower limit for z and z is the upper limit. From previous calculations we can write:

r ≤ z ≤2-r

The region E is then defined by:

Hence the integral becomes:

b) Integration in the order: drdzdθ. We adopt the following procedure 

• Fix a height z and angle θ.
• Integrate r first, over the region between the cone and paraboloid.
• Then integrate z and θ over their respective ranges.

 Let's determine the bounds:

• θ: The region is symmetric around the z-axis, so 0 ≤ θ ≤ 2π.
• z: At the tip of the cone, z = 0. At the top of the paraboloid, z = 2. So we have 0 ≤ z ≤ 2.
• r: For a fixed z and θ, r goes from the cone to the paraboloid:
  r = z (from z = r) to r = √(2 - z) (from z = 2 - r²).

 The region E is given by:

E = {(r, θ, z)/ z ≤ r≤⎷2 - z, 0≤z≤2, 0≤θ≤2ℼ}

The volume is given by:

.

Setting up a triple integral over a general region in cylindrical coordinates

 Goal: Set up a triple integral over a general region in cylindrical coordinates

If the cylindrical region over which we have to integrate is a solid, we project the region in one of the planes in the tri-dimensional space. Consequently, the triple integral of a function f(r, θ, z) over a region E is defined by:

Similar formulas exist for projections onto the other coordinate planes. Polar coordinates can be used in those planes if necessary.

Example

Consider the region E inside the right circular cylinder with equation r = 2 sinθ, bounded below by the rθ plane and above by the sphere of radius 4 centered at the origin. Set up the triple integral of the function f(r, θ,z) over the region E in cylindrical coordinates.

Solution

Let's set up the limits for z, r, θ.  The equation of a sphere in cylindrical coordinates is given by r² + z² = c² where c represents the radius of the sphere. Since the radius of the sphere is 4, we have  r² + z² = 16. Then  z = ⎷ 16 - r². Now we can set up the limits for z, r and θ. 

z varies from 0 ⎷16- r², r varies from 0 to 2 sinθ and θ varies from 0 to ℼ. The region E can be defined by:

Finally, the triple integral is defined by:

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:

.

 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

Average value of a function of three variables

 Goal: Find the average value of a function of three variables

The average value of a function of two variables is found by dividing the value of the double integral of the function over the region of the plane by the volume of that region. Similarly, the average value of a function of three variables is found by dividing the value of the triple integral of the function over the solid region by the volume of that region.

Theorem

If a function f(x,y,z) is integrable over a solid bounded region E with positive value V(E), then the average value of f is given by:

Example

Solution

Let's draw the region E:

The plane x + y + z =1 has the following intercepts with the axes: (1,0,0), (0,1,0) and (0,0,1). the region E is then determined by:

Practice