I Integral change of variables formula confusion

SchroedingersLion
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Greetings all.

I just got confused by the following.
Consider volume integral, for simplicity in 1D.
$$
V(A) = \int_{A} dz.
$$

If ##z## can be written as an invertible function of ##x##, i.e. ##z=f(x)##, we know the change of variables formula
$$
V(A)=\int_{A} dz= \int_{z^{-1}(A)} |z'(x)|dx.
$$

Intuitively, this is clear. What confused me now was the notion that the volume element transforms according to
$$
dz = |z'(x)|dx.
$$

If this equality holds, it should be allowed to write
$$
\int_{A} dz= \int_{A} |z'(x)|dx,
$$
which is obviously wrong. Why exactly is this not allowed?
 
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Are you wondering whether A or ##z^{-1}(A)## the domain of integration is ?
It would be A for ##\int dz ...## and ##z^{-1}(A)## for ##\int dx ...##.
 
Last edited:
SchroedingersLion said:
Greetings all.

I just got confused by the following.
Consider volume integral, for simplicity in 1D.
$$
V(A) = \int_{A} dz.
$$

If ##z## can be written as an invertible function of ##x##, i.e. ##z=f(x)##, we know the change of variables formula
$$
V(A)=\int_{A} dz= \int_{z^{-1}(A)} |z'(x)|dx.
$$

Intuitively, this is clear. What confused me now was the notion that the volume element transforms according to
$$
dz = |z'(x)|dx.
$$

If this equality holds, it should be allowed to write
$$
\int_{A} dz= \int_{A} |z'(x)|dx,
$$
which is obviously wrong. Why exactly is this not allowed?
Because written out, the integral reads ##\displaystyle{\int_{z=a}^{z=b}}dz##, and you cannot substitute the variable in one place and ignore the other places. Here is the multivariate version:
https://en.wikipedia.org/wiki/Integration_by_substitution#Substitution_for_multiple_variables
 
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fresh_42 said:
Because written out, the integral reads ##\displaystyle{\int_{z=a}^{z=b}}dz##, and you cannot substitute the variable in one place and ignore the other places. Here is the multivariate version:
https://en.wikipedia.org/wiki/Integration_by_substitution#Substitution_for_multiple_variables
Thank you. Writing the integration domain explicitly in terms of the variable helps to clear my confusion. I need to replace everything ##z##-related with the corresponding ##x## expression, not just a part of the whole thing.
 
Of course you have to change the integration range to the new variables. You also must make sure that the transformation is a diffeomorphism over the entire range of integration. For a 1D integral over an interval the function must be monotonous and differentiable verywhere along the interval. Then
$$\int_a^b \mathrm{d} z f(z) = \int_{z^{-1}(a)}^{z^{-1}(b)} \mathrm{d} u \frac{\mathrm{d}}{\mathrm{d}u} z(u)f[z(u)].$$
 
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