Multiple Integration

# 36 Change of Variables in Multiple Integrals

### Learning Objectives

- Determine the image of a region under a given transformation of variables.
- Compute the Jacobian of a given transformation.
- Evaluate a double integral using a change of variables.
- Evaluate a triple integral using a change of variables.

Recall from Substitution Rule the method of integration by substitution. When evaluating an integral such as we substitute Then or and the limits change to and Thus the integral becomes and this integral is much simpler to evaluate. In other words, when solving integration problems, we make appropriate substitutions to obtain an integral that becomes much simpler than the original integral.

We also used this idea when we transformed double integrals in rectangular coordinates to polar coordinates and transformed triple integrals in rectangular coordinates to cylindrical or spherical coordinates to make the computations simpler. More generally,

Where and and satisfy and

A similar result occurs in double integrals when we substitute and Then we get

where the domain is replaced by the domain in polar coordinates. Generally, the function that we use to change the variables to make the integration simpler is called a transformation or mapping.

### Planar Transformations

A planar transformation is a function that transforms a region in one plane into a region in another plane by a change of variables. Both and are subsets of For example, (Figure) shows a region in the transformed into a region in the by the change of variables and or sometimes we write and We shall typically assume that each of these functions has continuous first partial derivatives, which means and exist and are also continuous. The need for this requirement will become clear soon.

A transformation defined as is said to be a one-to-one transformation if no two points map to the same image point.

To show that is a one-to-one transformation, we assume and show that as a consequence we obtain If the transformation is one-to-one in the domain then the inverse exists with the domain such that and are identity functions.

(Figure) shows the mapping where and are related to and by the equations and The region is the domain of and the region is the range of also known as the *image* of under the transformation

Suppose a transformation is defined as where Find the image of the polar rectangle in the to a region in the Show that is a one-to-one transformation in and find

Since varies from 0 to 1 in the we have a circular disc of radius 0 to 1 in the Because varies from 0 to in the we end up getting a quarter circle of radius in the first quadrant of the ((Figure)). Hence is a quarter circle bounded by in the first quadrant.

In order to show that is a one-to-one transformation, assume and show as a consequence that In this case, we have

Dividing, we obtain

since the tangent function is one-one function in the interval Also, since we have Therefore, and is a one-to-one transformation from into

To find solve for in terms of We already know that and Thus is defined as and

Let the transformation be defined by where and Find the image of the triangle in the with vertices and

The triangle and its image are shown in (Figure). To understand how the sides of the triangle transform, call the side that joins and side the side that joins and side and the side that joins and side

For the side transforms to so this is the side that joins and

For the side transforms to so this is the side that joins and

For the side transforms to (hence so this is the side that makes the upper half of the parabolic arc joining and

All the points in the entire region of the triangle in the are mapped inside the parabolic region in the

Let a transformation be defined as where Find the image of the rectangle from the after the transformation into a region in the Show that is a one-to-one transformation and find

where and

Follow the steps of (Figure).

### Jacobians

Recall that we mentioned near the beginning of this section that each of the component functions must have continuous first partial derivatives, which means that and exist and are also continuous. A transformation that has this property is called a transformation (here denotes continuous). Let where and be a one-to-one transformation. We want to see how it transforms a small rectangular region units by units, in the (see the following figure).

Since and we have the position vector of the image of the point Suppose that is the coordinate of the point at the lower left corner that mapped to The line maps to the image curve with vector function and the tangent vector at to the image curve is

Similarly, the line maps to the image curve with vector function and the tangent vector at to the image curve is

Now, note that

Similarly,

This allows us to estimate the area of the image by finding the area of the parallelogram formed by the sides and By using the cross product of these two vectors by adding the **k**th component as the area of the image (refer to The Cross Product) is approximately In determinant form, the cross product is

Since we have

The Jacobian of the transformation is denoted by and is defined by the determinant

Using the definition, we have

Note that the Jacobian is frequently denoted simply by

Note also that

Hence the notation suggests that we can write the Jacobian determinant with partials of in the first row and partials of in the second row.

Find the Jacobian of the transformation given in (Figure).

The transformation in the example is where and Thus the Jacobian is

Find the Jacobian of the transformation given in (Figure).

The transformation in the example is where and Thus the Jacobian is

Find the Jacobian of the transformation given in the previous checkpoint:

Follow the steps in the previous two examples.

### Change of Variables for Double Integrals

We have already seen that, under the change of variables where and a small region in the is related to the area formed by the product in the by the approximation

Now let’s go back to the definition of double integral for a minute:

Referring to (Figure), observe that we divided the region in the into small subrectangles and we let the subrectangles in the be the images of under the transformation

Then the double integral becomes

Notice this is exactly the double Riemann sum for the integral

Let where and be a one-to-one transformation, with a nonzero Jacobian on the interior of the region in the it maps into the region in the If is continuous on then

With this theorem for double integrals, we can change the variables from to in a double integral simply by replacing

when we use the substitutions and and then change the limits of integration accordingly. This change of variables often makes any computations much simpler.

Consider the integral

Use the change of variables and and find the resulting integral.

First we need to find the region of integration. This region is bounded below by and above by (see the following figure).

Squaring and collecting terms, we find that the region is the upper half of the circle that is, In polar coordinates, the circle is so the region of integration in polar coordinates is bounded by and

The Jacobian is as shown in (Figure). Since we have

The integrand changes to in polar coordinates, so the double iterated integral is

Considering the integral use the change of variables and and find the resulting integral.

Follow the steps in the previous example.

Notice in the next example that the region over which we are to integrate may suggest a suitable transformation for the integration. This is a common and important situation.

Consider the integral where is the parallelogram joining the points and ((Figure)). Make appropriate changes of variables, and write the resulting integral.

First, we need to understand the region over which we are to integrate. The sides of the parallelogram are ((Figure)). Another way to look at them is and

Clearly the parallelogram is bounded by the lines and

Notice that if we were to make and then the limits on the integral would be and

To solve for and we multiply the first equation by and subtract the second equation, Then we have Moreover, if we simply subtract the second equation from the first, we get and

Thus, we can choose the transformation

and compute the Jacobian We have

Therefore, Also, the original integrand becomes

Therefore, by the use of the transformation the integral changes to

which is much simpler to compute.

Make appropriate changes of variables in the integral where is the trapezoid bounded by the lines Write the resulting integral.

and and

Follow the steps in the previous example.

We are ready to give a problem-solving strategy for change of variables.

- Sketch the region given by the problem in the and then write the equations of the curves that form the boundary.
- Depending on the region or the integrand, choose the transformations and
- Determine the new limits of integration in the
- Find the Jacobian
- In the integrand, replace the variables to obtain the new integrand.
- Replace or whichever occurs, by

In the next example, we find a substitution that makes the integrand much simpler to compute.

Using the change of variables and evaluate the integral

where is the region bounded by the lines and and the curves and (see the first region in (Figure)).

As before, first find the region and picture the transformation so it becomes easier to obtain the limits of integration after the transformations are made ((Figure)).

Given and we have and and hence the transformation to use is The lines and become and respectively. The curves and become and respectively.

Thus we can describe the region (see the second region (Figure)) as

The Jacobian for this transformation is

Therefore, by using the transformation the integral changes to

Doing the evaluation, we have

Using the substitutions and evaluate the integral where is the region bounded by the lines

Sketch a picture and find the limits of integration.

### Change of Variables for Triple Integrals

Changing variables in triple integrals works in exactly the same way. Cylindrical and spherical coordinate substitutions are special cases of this method, which we demonstrate here.

Suppose that is a region in and is mapped to in ((Figure)) by a one-to-one transformation where and

Then any function defined on can be thought of as another function that is defined on

Now we need to define the Jacobian for three variables.

The Jacobian determinant in three variables is defined as follows:

This is also the same as

The Jacobian can also be simply denoted as

With the transformations and the Jacobian for three variables, we are ready to establish the theorem that describes change of variables for triple integrals.

Let where and be a one-to-one transformation, with a nonzero Jacobian, that maps the region in the into the region in the As in the two-dimensional case, if is continuous on then

Let us now see how changes in triple integrals for cylindrical and spherical coordinates are affected by this theorem. We expect to obtain the same formulas as in Triple Integrals in Cylindrical and Spherical Coordinates.

Derive the formula in triple integrals for

- cylindrical and
- spherical coordinates.

- For cylindrical coordinates, the transformation is from the Cartesian to the Cartesian ((Figure)). Here and The Jacobian for the transformation is

We know that so Then the triple integral is

- For spherical coordinates, the transformation is from the Cartesian to the Cartesian ((Figure)). Here and The Jacobian for the transformation is

Expanding the determinant with respect to the third row:

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Since we must have Thus

Then the triple integral becomes

Let’s try another example with a different substitution.

Evaluate the triple integral

in by using the transformation

Then integrate over an appropriate region in

As before, some kind of sketch of the region in over which we have to perform the integration can help identify the region in ((Figure)). Clearly in is bounded by the planes We also know that we have to use for the transformations. We need to solve for Here we find that and

Using elementary algebra, we can find the corresponding surfaces for the region and the limits of integration in It is convenient to list these equations in a table.

Equations in for the region | Corresponding equations in for the region | Limits for the integration in |
---|---|---|

Now we can calculate the Jacobian for the transformation:

The function to be integrated becomes

We are now ready to put everything together and complete the problem.

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Let be the region in defined by

Evaluate by using the transformation and

Make a table for each surface of the regions and decide on the limits, as shown in the example.

### Key Concepts

- A transformation is a function that transforms a region in one plane (space) into a region in another plane (space) by a change of variables.
- A transformation defined as is said to be a one-to-one transformation if no two points map to the same image point.
- If is continuous on then
- If is continuous on then

In the following exercises, the function on the region bounded by the unit square is given, where is the image of under

- Justify that the function is a transformation.
- Find the images of the vertices of the unit square through the function
- Determine the image of the unit square and graph it.

a. and The functions and are continuous and differentiable, and the partial derivatives are continuous on b. and c. is the rectangle of vertices in the the following figure.

a. and The functions and are continuous and differentiable, and the partial derivatives and are continuous on b. and c. is the parallelogram of vertices in the see the following figure.

a. and The functions and are continuous and differentiable, and the partial derivatives and are continuous on b. and c. is the unit square in the see the figure in the answer to the previous exercise.

In the following exercises, determine whether the transformations are one-to-one or not.

is the rectangle of vertices

is the triangle of vertices

is not one-to-one: two points of have the same image. Indeed,

is the square of vertices

where is the triangle of vertices

is one-to-one: We argue by contradiction. implies and Thus, and

where

where

is not one-to-one:

In the following exercises, the transformations are one-to-one. Find their related inverse transformations

where

where

where and

where and

where

where

In the following exercises, the transformation and the region are given. Find the region

where

where

where

where

In the following exercises, find the Jacobian of the transformation.

The triangular region with the vertices is shown in the following figure.

- Find a transformation where and are real numbers with such that and
- Use the transformation to find the area of the region

The triangular region with the vertices is shown in the following figure.

- Find a transformation where and are real numbers with such that and
- Use the transformation to find the area of the region

a. b. The area of is

In the following exercises, use the transformation to evaluate the integrals on the parallelogram of vertices shown in the following figure.

In the following exercises, use the transformation to evaluate the integrals on the square determined by the lines and shown in the following figure.

In the following exercises, use the transformation to evaluate the integrals on the region bounded by the ellipse shown in the following figure.

In the following exercises, use the transformation to evaluate the integrals on the trapezoidal region determined by the points shown in the following figure.

The circular annulus sector bounded by the circles and the line and the is shown in the following figure. Find a transformation from a rectangular region in the to the region in the Graph

The solid bounded by the circular cylinder and the planes is shown in the following figure. Find a transformation from a cylindrical box in to the solid in

in the

Show that where is a continuous function on and is the region bounded by the ellipse

Show that where is a continuous function on and is the region bounded by the ellipsoid

**[T]** Find the area of the region bounded by the curves and by using the transformation and Use a computer algebra system (CAS) to graph the boundary curves of the region

**[T]** Find the area of the region bounded by the curves and by using the transformation and Use a CAS to graph the boundary curves of the region

The area of is the boundary curves of are graphed in the following figure.

Evaluate the triple integral by using the transformation

Evaluate the triple integral by using the transformation

A transformation of the form where are real numbers, is called linear. Show that a linear transformation for which maps parallelograms to parallelograms.

The transformation where is called a rotation of angle Show that the inverse transformation of satisfies where is the rotation of angle

**[T]** Find the region in the whose image through a rotation of angle is the region enclosed by the ellipse Use a CAS to answer the following questions.

- Graph the region
- Evaluate the integral Round your answer to two decimal places.

**[T]** The transformations defined by and are called reflections about the origin, and the line respectively.

- Find the image of the region in the through the transformation
- Use a CAS to graph
- Evaluate the integral by using a CAS. Round your answer to two decimal places.

a. b. is graphed in the following figure;

c.

**[T]** The transformation of the form where is a positive real number, is called a stretch if and a compression if in the Use a CAS to evaluate the integral on the solid by considering the compression defined by and Round your answer to four decimal places.

**[T]** The transformation where is a real number, is called a shear in the The transformation, where is a real number, is called a shear in the

- Find transformations
- Find the image of the trapezoidal region bounded by and through the transformation
- Use a CAS to graph the image in the
- Find the area of the region by using the area of region

a. b. The image is the quadrilateral of vertices c. is graphed in the following figure;

d.

Use the transformation, and spherical coordinates to show that the volume of a region bounded by the spheroid is

Find the volume of a football whose shape is a spheroid whose length from tip to tip is inches and circumference at the center is inches. Round your answer to two decimal places.

**[T]** Lamé ovals (or superellipses) are plane curves of equations where *a*, *b*, and *n* are positive real numbers.

- Use a CAS to graph the regions bounded by Lamé ovals for and respectively.
- Find the transformations that map the region bounded by the Lamé oval also called a squircle and graphed in the following figure, into the unit disk.

- Use a CAS to find an approximation of the area of the region bounded by Round your answer to two decimal places.

**[T]** Lamé ovals have been consistently used by designers and architects. For instance, Gerald Robinson, a Canadian architect, has designed a parking garage in a shopping center in Peterborough, Ontario, in the shape of a superellipse of the equation with and Use a CAS to find an approximation of the area of the parking garage in the case yards, yards, and yards.

### Chapter Review Exercises

*True or False?* Justify your answer with a proof or a counterexample.

Fubini’s theorem can be extended to three dimensions, as long as is continuous in all variables.

True.

The integral represents the volume of a right cone.

The Jacobian of the transformation for is given by

False.

Evaluate the following integrals.

0

where is a disk of radius centered at the origin

where

1.475

For the following problems, find the specified area or volume.

The area of region enclosed by one petal of

The volume of the solid that lies between the paraboloid and the plane

The volume of the solid bounded by the cylinder and from to

93.291

The volume of the intersection between two spheres of radius 1, the top whose center is and the bottom, which is centered at

For the following problems, find the center of mass of the region.

on the circle with radius in the first quadrant only.

in the region bounded by and

on the inverted cone with radius and height

The volume an ice cream cone that is given by the solid above and below

The following problems examine Mount Holly in the state of Michigan. Mount Holly is a landfill that was converted into a ski resort. The shape of Mount Holly can be approximated by a right circular cone of height ft and radius ft.

If the compacted trash used to build Mount Holly on average has a density find the amount of work required to build the mountain.

ft-lb

In reality, it is very likely that the trash at the bottom of Mount Holly has become more compacted with all the weight of the above trash. Consider a density function with respect to height: the density at the top of the mountain is still density and the density increases. Every feet deeper, the density doubles. What is the total weight of Mount Holly?

The following problems consider the temperature and density of Earth’s layers.

**[T]** The temperature of Earth’s layers is exhibited in the table below. Use your calculator to fit a polynomial of degree to the temperature along the radius of the Earth. Then find the average temperature of Earth. (*Hint*: begin at in the inner core and increase outward toward the surface)

Layer | Depth from center (km) | Temperature |
---|---|---|

Rocky Crust | 0 to 40 | 0 |

Upper Mantle | 40 to 150 | 870 |

Mantle | 400 to 650 | 870 |

Inner Mantel | 650 to 2700 | 870 |

Molten Outer Core | 2890 to 5150 | 4300 |

Inner Core | 5150 to 6378 | 7200 |

average temperature approximately

**[T]** The density of Earth’s layers is displayed in the table below. Using your calculator or a computer program, find the best-fit quadratic equation to the density. Using this equation, find the total mass of Earth.

Layer | Depth from center (km) | Density (g/cm3) |
---|---|---|

Inner Core | ||

Outer Core | ||

Mantle | ||

Upper Mantle | ||

Crust |

The following problems concern the Theorem of Pappus (see Moments and Centers of Mass for a refresher), a method for calculating volume using centroids. Assuming a region when you revolve around the the volume is given by and when you revolve around the the volume is given by where is the area of Consider the region bounded by and above

Find the volume when you revolve the region around the

Find the volume when you revolve the region around the

### Glossary

- Jacobian
- the Jacobian in two variables is a determinant:

the Jacobian in three variables is a determinant:

- one-to-one transformation
- a transformation defined as is said to be one-to-one if no two points map to the same image point

- planar transformation
- a function that transforms a region in one plane into a region in another plane by a change of variables

- transformation
- a function that transforms a region in one plane into a region in another plane by a change of variables