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

Changing the order of integration and coordinate systems

As we have seen previously, in order to change the order of integration of a triple integral of a function over a bounded region, the region has to be projected on one of the three plane coordinate systems of the three-dimensional space coordinates. In this post we learned to project the region E in a different coordinates system when the triple integral becomes difficult to calculate. As the double integral seems to be difficult to calculate, we change from plane to polar coordinates to make the computation easier.

Example

y = x² + z² and the plane y = 4

Solution

In order to define the region E, let's project it on the xy plane. The projection of E onto this plane is the region bounded above by above by the plane y = 4 and below by the parabola y = x² This is shown in the following figure:

According to the projection above, the region E is then determined by:

Changing the order of integration in a triple integral

 Goal: Change the order of integration in a triple integral over a  general bounded region.

Considerations

Changing the order of integration in double integrals facilitates their computations. In triple integrals over a rectangular box, changing the order of integration doesn't simplify the calculations. However, changing the order of integration in a triple integral over a general bounded region facilitates the computation.

Example

Solution

Let's start by defining the region E and the form in which the triple integral should be expressed once the order of integration is changed:

Let's draw the projections on the three coordinates plane:

Let's redefine E for the change of the order of integration:

The integral becomes:

Let's calculate the given integral by substituting f(x,y,z) by xyz.

= 1/168

Let's calculate the second integral with the order of integration changed.

The answers for both integrals are identical.

Finding a volume by evaluating triple integrals

 Goal: Find the volume of a solid using a triple integral.

Just as we used a double integral to find an area, we are going to use a triple integral to find a volume.

In the case of an area, we used the formula:

In the case of the volume, we are going to use:

Let's do an example:

Example

Solution

The volume of the pyramid is then 4/3 cubic feet.