Download The dangers of blindly changing variables

CHANGE OF VARIABLES
JOHN THICKSTUN
Let f : R → R be the simple quadratic function f (x) = x2 . The fundamental theorem
of calculus tells us that
Z 1
1 3 1
2
1 1
2
x dx = x = + = .
3 −1 3 3
3
−1
Alternatively, we can make a naive substitution u = x2 . Some formal manipulations give
du
√
us du = 2xdx and therefore dx = du
2x = 2 u . Changing the limits of integration, we have
Z
Z 1
Z 1
u
1 1√
2
√ du =
udu.
x dx =
2 1
−1
1 2 u
Integrating over a single point is just zero, which implies that the right-hand side of the
above integral vanishes. But our earlier calculation shows that this is clearly not true;
something has gone wrong. The problem is that our substituting function is not injective.
Let’s try again, this time splitting our integral into two segments for which u is injective
before making the u-substitution:
Z 1
Z 0
Z 1
Z
Z
1 0√
1 1√
x2 dx =
x2 dx +
x2 dx =
udu +
udu.
2 1
2 0
−1
−1
0
Z
Z
1 1√
1 1√
udu +
udu = 0.
=−
2 0
2 0
So we’re still in trouble. Now the problem is that we took the positive square root of u.
That’s ok, in fact the general change of variables formula tells us to do this, but we need
to be careful about the orientation of our domain. So let’s back up again and clarify our
domain of integration
Z 1
Z
Z
Z
2
2
2
x dx =
x dx =
x dx +
x2 dx
−1
[−1,1]
√
[−1,0)
Z
1
udu +
2
Z
=
√
[0,1]
Z
√
√
1
1
udu =
udu +
udu
2 (0,1]
2 [0,1]
u([−1,0))
u([0,1])
1√
2 3/2 1 2
udu = u = .
3
3
0
0
Stripping the orientation from our domain of integration cancels with taking the positive
square root, so now we are good!
1
=
2
Z
Z
1
2
JOHN THICKSTUN
Let’s have some more fun. Instead of integrating over [−1, 1], which is pretty boring,
let’s integrate over the half-circle C = {eiθ ∈ C; 0 ≤ θ ≤ π}. We can pull this curve back
to R with the parameterization c(θ) = eiθ where θ ∈ [0, π] and write
Z
Z
Z
Z
2 0
2
2
e2iθ eiθ dθ
c(θ) c (θ)dθ = i
x dx = x dx =
C
[0,π]
c
[0,π]
i 3iθ 0
2
3iθ
e dθ = e = .
=i
3i
3
[0,π]
pi
Of course, we could have predicted this result from Cauchy’s integral theorem.
Z