The 'integration by change of variable' theorem

[quasar987]

In my analysis textbook, the 'integration by change of variable' theorem reads

"Theorem: Consider f a continuous fonction on [a,b], g a continuous function on [c,d] such that g' is continuous on [c,d]. If g([c,d]) is a subset of [a,b] and if g(c) = a and g(d) = b, then

$$\int_a^b f(x)dx = \int_c^d f(g(t))g'(t)dt$$


But reading the proof, I nowhere see the need for g([c,d]) being a subset of [a,b]. Is this really necessary? As long as g(c) = a and g(d) = b, it should be alright, no?

 

[HallsofIvy]

No. Suppose a is 0, b= 1, and g(x) is the function g(x)= 8x(1-x)= 8x- 8x². To make it simple, suppose f is the constant function f(x)= 1.

Then g'(x)= 8- 16x and the theorem, without that requirement, would assert that
$$\int_0^1dx= 1= \int_0^1(8- 16x)dx= 0!$$

 

[quasar987]

But you use [0,1] has your interval [c,d]. And while g(0) = 0, g(1) does not equal 1. It's 0 too.

 

[philosophking]

*bump*

Yes, I'm still curious about this also. It looks like the definition of a line integral in complex analysis:

If

$$\int_{\gamma}f(z)dz=\int_a^bf(g(t))g'(t)dt$$

...g(t) is the function for the parametrization of the curve.

right? And in this case, we know that the parametrization does not matter: as long as you start and end at the correct points.

(Also, forgive my LaTeX!)

 

[quasar987]

Mathworld does not mention the need for g([c,d]) to be a subset of [a,b] in its statement of change of variable theorem:


http://mathworld.wolfram.com/ChangeofVariablesTheorem.html

 

[snoble]

It makes sense that it shouldn't matter. Think of a very simple case where g(x) on [c,d] has a single maximal point at c<m<d and say g(c)<g(d)<g(m). So there is a c1, c< c1 < m such that g(c1)=g(d). So since this is a simple case we'll say g([c,c1]) is a subset of [g(c),g(d)]. So
$$\int_{g(c)}^{g(d)} f(x) dx = \int_c^{c1} g'(x)g(x)dx$$ and
$$\int_{c1}^m g'(x)g(x)dx = \int_{g(c1)}^{g(m)} f(x) dx = -\int_{g(m)}^{g(c1)} f(x) dx=- \int_{m}^d g'(x)g(x)dx$$
So the sum cancels out nicely.

Remember this is a simplified case and in no way am I saying this is a proof for the general case. You can have infinite oscillations so that you are dealing with an infinite sum with no guarantee (from me at least) that it will converge where you want it to.