Section 13.11: Change of Variables in Multiple Integrals A change of

Section 13.11: Change of Variables in Multiple Integrals
A change of variables can be useful when evaluating double or triple integrals. For
instance, changing from Cartesian coordinates to polar coordinates is often useful. A change
of variables can usually be described by a transformation.
Definition: A transformation T from the uv-plane to the xy-plane is a function which maps
points (u, v) to points (x, y) according to the rule:
T (u, v) = (x(u, v), y(u, v)) = (x, y).
A transformation T maps a region S in the uv-plane to a region R in the xy-plane called
the image of S. That is,
R = {T (u, v)|(u, v) ∈ S}.
We will usually work with C 1 transformations which are transformations whose component
functions x(u, v) and y(u, v) have continuous first-order partial derivatives.
Example: Find the image of the set S = {(u, v)|0 ≤ u ≤ 2, 0 ≤ v ≤ 1} under the transformation T defined by the equations x = u − 2v and y = 2u − v.
The rectangular region S = [0, 2] × [0, 1] in the uv-plane is mapped to the parallelogram in
the xy-plane with vertices (0, 0), (2, 4), (−2, −1), and (0, 3).
Definition: The Jacobian of the transformation T defined by x = x(u, v) and y = y(u, v) is
∂x ∂x ∂(x, y) ∂u ∂v ∂x ∂y ∂x ∂y
=
−
.
=
∂(u, v) ∂y ∂y ∂u ∂v ∂v ∂u
∂u ∂v
Example: Find the Jacobian of the transformation defined by x = r cos θ and y = r sin θ.
The Jacobian of the transformation is
∂(x, y) cos θ −r sin θ
=
sin θ r cos θ
∂(r, θ)
= r(cos2 θ + sin2 θ) = r.
Theorem: (Change of Variables in a Double Integral)
Suppose that T is a one-to-one C 1 transformation that maps a region S in the uv-plane to
a region R in the xy-plane and whose Jacobian is nonzero. Suppose that f is continuous on
R and that R and S are type I or type II regions. Then
ZZ
ZZ
∂(x, y) dudv.
f (x, y)dxdy =
f (x(u, v), y(u, v)) ∂(u, v) R
S
ZZ
x2 dA, where R is
Example: Use the change of variables x = 2u and y = 3v to evaluate
R
the region bounded by the ellipse 9x2 + 4y 2 = 36.
The given transformation maps the ellipse to a circle:
9x2 + 4y 2 = 36
x2 y 2
+
= 1
4
9
u2 + v 2 = 1.
The Jacobian of this transformation is
∂(x, y) 2 0
=
∂(u, v) 0 3
= 6.
By the previous theorem,
ZZ
ZZ
2
4u2 dA
Z 2π Z 1
r3 cos2 θdrdθ
24
0
0
Z 2π
cos2 θdθ
6
Z0 2π
(1 + cos 2θ)dθ
3
0
2π
1
3 θ + sin 2θ 2
0
6π.
x dA = 6
R
=
=
=
=
=
Example: Use the transformation defined by x =
u
v
ZZ
and y = v to evaluate
xydA, where R
R
is the region in the first quadrant bounded by the lines y = x and y = 3x and the hyperbolas
xy = 1 and xy = 3.
The given transformation maps the hyperbolas xy = 1 and xy = 3√to the lines u√= 1 and
u = 3 and maps the lines y = x and y = 3x to the parabolas v = u and v = 3u. The
Jacobian of this transformation is
1
u ∂(x, y) − 2 1
= v
v = .
∂(u, v) 0
1 v
By the previous theorem,
ZZ
ZZ
xydA =
R
=
=
=
=
=
u
dA
S v
√
Z 3 Z 3u
u
dvdu
√
v
1
u
Z 3 h √
√ i
u ln( 3u) − ln( u) du
1
Z
1 3
ln(3)udu
2 1
3
ln(3) 2 u
4
1
2 ln(3).
RR
y−x
Example: Make an appropriate change of variables to evaluate R 5 cos 9 y+x
dA, where R
is the trapezoidal region with vertices (2, 0), (5, 0), (0, 5), and (0, 2).
Let u = y − x and v = y + x. Then x = 12 (v − u) and y = 21 (u + v). This transformation
maps the vertices of R to (−2, 2), (−5, 5), (0, 5), and (2, 2). The region S can be described
as
S = {(u, v)|2 ≤ v ≤ 5, −v ≤ u ≤ v}.
The Jacobian of the transformation is
1
∂(x, y) − 2
=
∂(u, v) 1
2
1
2
1
2
1
=− .
2
By the Theorem,
ZZ
ZZ
y−x
9u
1
5 cos 9
cos
dA = 5
dA
y+x
v
2
R
S
Z Z
9u
5 5 v
cos
=
dudv
2 2 −v
v
v
Z 5
5
9u =
sin
dv
18 2
v −v
Z 5
5
=
v(sin(9) − sin(−9))dv
18 2
Z 5
5
vdv
=
sin(9)
9
2
5
=
sin(9)(25 − 4)
18
35
=
sin(9).
6
There is a similar change of variables formula for triple integrals. Consider a C 1 transformation T that maps a region S in uvw-space to a region R in xyz-space according to the
rule:
T (u, v, w) = (x(u, v, w), y(u, v, w), z(u, v, w)) = (x, y, z).
The Jacobian of this transformation is
∂(x, y, z)
= ∂(u, v, w) ∂x
∂u
∂y
∂u
∂z
∂u
∂x
∂v
∂y
∂v
∂z
∂v
∂x
∂w
∂y
∂w
∂z
∂w
.
Theorem: (Change of Variables in a Triple Integral)
Suppose that T is a one-to-one C 1 transformation that maps a region S in uvw-space to a
region R in xyz-space and whose Jacobian is nonzero. Suppose that f is continuous on R
and that R and S are type 1, 2, or 3 regions. Then
ZZZ
ZZZ
∂(x, y, z) dudvdw.
f (x, y, z)dxdydz =
f (x(u, v, w), y(u, v, w), z(u, v, w)) ∂(u, v, w) R
S

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