# Change of Variables in Multiple Integrals

```Calc 3 Lecture Notes
Section 13.8
Page 1 of 11
Section 13.8: Change of Variables in Multiple Integrals
Big idea: Changing variables can simplify the integration boundaries and integrands of many multiple integrals.
In this section, we learn a general formalism for transforming integrals using transformations besides cylindrical
or spherical.
Big skill: You should be able to find a transformation that simplifies a multiple integral.
Note the following integral transformation (from section 13.3):
1 1 x2
 
0
0
 x2  y 2 
3/ 2
 /2 1
dydx 
  r 
2 3/ 2
rdrd
0 0
Three aspects of the integral had to be transformed:
1. The integrand was transformed using the transformations x  r cos and y  r sin  .
2. The integration region was transformed from a quarter-circle in the x-y plane to a rectangle in the “r- ”
plane:
3. The differential area element was transformed from dxdy to rdrd by geometrically analyzing small
area elements in the x-y plane. Partitions of the x-y plane had the shape of annular sectors, while
corresponding partitions of the r- plane are rectangular:
In this section, we will be responsible for finding a variable transformation that simplifies the integrand and/or
the region of integration. The book provides a formula for transforming the differential area or volume element
once the transformation has been chosen.
A little formalism before we look at making some transformations:
A transformation T from the u-v plane to the x-y plane is a function that maps points in the u-v plane to points
in the x-y plane. The shorthand notation is:
Calc 3 Lecture Notes
T  u, v    x, y 
Section 13.8
Page 2 of 11
where
x  g  u, v  and y  h  u, v 
for some functions g and h.

A change of variables for a double integral is defined by a transformation T from a region S in the u-v
plane to a region R in the x-y plane.

R is called the image of S under the transformation T.

T is one-to-one on S if for every point (x, y) in R there is exactly one point (u, v) in S such that T(u, v) =
(x, y).
o This implies we can solve for u and v in terms of x and y.

Also, we will restrict our transformations to those where g and h have continuous first partial derivatives
in S.
Calc 3 Lecture Notes
Section 13.8
Page 3 of 11
Practice:
1. Let R be the region bounded by the lines y  2 x , y  2 x  4 , y  0.5 x , and y  0.5 x  2 . Find a
transformation T that maps a rectangular region in the u-v plane onto this parallelogram. Notice that you
can show that the intersection points are (0, 0), (4/3, 8/3), (4, 4), and (8/3, 4/3). Show how the
boundaries of a double integral are simplified by the transformation.
Calc 3 Lecture Notes
Section 13.8
Page 4 of 11
1
2
1
Let R be the region bounded by the hyperbolae y  and y  and the lines y  2 x and y  x . Find a
x
x
2
transformation T that maps a rectangular region in the u-v plane. Show how the boundaries of a double integral
are simplified by the transformation.
Calc 3 Lecture Notes
Section 13.8
Page 5 of 11
Now that we’ve had some practice finding transformations, let’s derive a formula for computing the
differential area element given a transformation.
First suppose that we have a transformation from (u, v) onto (x, y). Under this transformation, rectangular
partitions of the region S will transform to non-rectangular partitions of the region R.
The problem is, we need the areas Ai of each of the curvilinear regions Ri, because those areas are used in
computing double integrals:

R
n
f  x, y  dA   f  xi , yi  Ai
i 1
The trick is to approximate each Ai as a parallelogram whose four corners come from the transformed
coordinates of the rectilinear regions Si:
The points A, B, C, and D have coordinates determined by the transformation T:
A   xi , yi    g  ui , vi  , h  ui , vi  
B   g  ui  u , vi  , h  ui  u , vi  
C   g  ui  u , vi  v  , h  ui  u , vi  v  
D   g  ui , vi  v  , h  ui , vi  v  
Calc 3 Lecture Notes
Section 13.8
Page 6 of 11
These coordinates can be used to compute the vectors AB and AD . We want these vectors because the
area of the parallelogram they describe can be computed using AB  AD .
AB  g  ui  u , vi   g  ui , vi  , h  ui  u , vi   h  ui , vi 
AD  g  ui , vi  v   g  ui , vi  , h  ui , vi  v   h  ui , vi 
From the definition of partial derivatives,
So, for u and v small,
g  ui  u, vi   g  ui , vi 
g
.
 ui , vi   lim
u 0
u
u
g
 ui , vi  u
u
h
h  ui  u, vi   h  ui , vi  
 ui , vi  u
u
g
g  ui , vi  v   g  ui , vi  
 ui , vi  v
v
h
h  ui , vi  v   h  ui , vi    ui , vi  v
v
g  ui  u, vi   g  ui , vi  
So, our vectors simplify to:
g
h
AB 
 ui , vi  ,  ui , vi  u
u
u
g
h
 ui , vi  ,  ui , vi  v
v
v
And now we can compute the cross section:
i
j
k
g
h
 ui , vi  u
 ui , vi  u 0
u
u
g
h
 ui , vi  v
 ui , vi  v 0
v
v
g
h
 ui , vi 
 ui , vi 
u
u

u vk
g
h
 ui , vi 
 ui , vi 
v
v
And now we can compute area:
g
h
 ui , vi 
 ui , vi 
u
u
Ai  AB  AD 
uv 
g
h
 ui , vi 
 ui , vi 
v
v
g
g
 ui , vi 
 ui , vi 
u
v
uv 
h
h
 ui , vi 
 ui , vi 
u
v
x
x
 ui , vi 
 ui , vi 
u
v
uv
y
y
 ui , vi 
 ui , vi 
u
v
Calc 3 Lecture Notes
Section 13.8
Page 7 of 11
Definition 8.1: Jacobian of a Transformation
The determinant
x x
u v
y y
u v
is called the Jacobian of a transformation T and is written using the notation
  x, y 
  u, v 
Given this area transformation and definition, we can convert our integral to the u-v plane:

n
f  x, y  dA   f  xi , yi  Ai
i 1
R
n
  f  xi , yi 
i 1
  x, y 
u v
  u, v 
n
  f  g  ui , vi  , h  ui , vi  
i 1
  f  g  u , v  , h  u , v  
S
  x, y 
  u, v 
  x, y 
  u, v 
u v
dudv
Theorem 8.1: Change of Variables in Double Integrals
If a region S in the u-v plane is mapped onto the region R in the x-y plane by the one-to-one transformation T
defined by x  g  u, v  and y  h  u, v  , where g and h have continuous first derivatives on S, and if f is
continuous on R and the Jacobian

R
  x, y 
  u, v 
f  x, y  dA   f  g  u , v  , h  u , v  
S
is nonzero on S, then
  x, y 
  u, v 
dudv .
Calc 3 Lecture Notes
Section 13.8
Page 8 of 11
Practice:
2. Show that the Jacobian yields the correct differential area element dA for polar coordinates.
3. Compute the Jacobian for the hyperbolic transformation from page 4.
Calc 3 Lecture Notes
4. Compute
Section 13.8
Page 9 of 11
 8  x  2 y  dA for the region R shown below first using Cartesian coordinates and then
R
using transformed coordinates.
Calc 3 Lecture Notes
Section 13.8
Page 10 of 11
Change of variables for triple integrals:
In three dimensions, a change of variables is fairly analogous to the two dimensional case:
Given a transformation T from a region S of u-v-w space onto a region R of x-y-z space, specified by the
functions
x  g  u, v, w , y  h  u, v, w , and z   u, v, w ,
the Jacobian is defined as:
x x x
u v w
  x, y, z  y y y

  u , v, w  u v w
z z z
u v w
Theorem 8.2: Change of Variables in Triple Integrals
If a region S in u-v-w space is mapped onto the region R in x-y-z space by the one-to-one transformation T
defined by x  g  u, v, w , y  h  u, v, w , and z   u, v, w , where g, h, and have continuous first derivatives
on S, and if f is continuous on R and the Jacobian

R
  x, y , z 
  u , v, w 
f  x, y, z  dV   f  g  u , v, w  , h  u , v, w  ,
S
is nonzero on S, then
 u , v, w  
  x, y , z 
  u , v, w 
dudvdw .
Practice:
5. Show that the Jacobian yields the correct differential volume element dV for spherical coordinates.
Calc 3 Lecture Notes
Section 13.8
Page 11 of 11
6. Toroidal coordinates are used to specify the location of points inside toroids as shown below. The
center of the toroid is a circle of radius a (called the major radius) in the x-y plane, and points are located
by an angle  measured from the standard position in the x-y plane, a distance r measured from the
major circle, and an angle  measured in the plane  = k. The transformation is specified by:
x   a  r cos   cos
y   a  r cos   sin 
z  r sin 
Compute the volume of a torus with a major radius of 2 and a minor radius of 1. Compare this to
the answer you get from Pappus’ Second Theorem, which says that the volume of a solid of revolution
equals the cross-sectional area of the rotated lamina times the distance the traveled by its centroid.
```