1

To give you a bit of context, if I'm trying to calculate the work done by a force from a distance $x_0$ (given constant) to $x$ (variable), I've been told (in lectures) to write the work done as $$W=\int_{x_0}^x F(\xi)d\xi \tag{1},$$ rather than $$W=\int_{x_0}^xF(x)dx \tag{2},$$ the reason being that "we've got the same variable in two places".

But what exactly is wrong with this?

Is $(2)$ wrong?

beep-boop
  • 11,595
  • 1
    Technically there is nothing "wrong" with it. However, it leads to confusion - as the $x$ in the integrand is not the same as the $x$ in the limit. It's best avoided – Mathmo123 Jun 30 '14 at 21:55
  • 1
    See : http://math.stackexchange.com/questions/109105/limit-of-integration-cant-be-the-same-as-variable-of-integration?rq=1 – user99680 Jun 30 '14 at 21:57
  • Technically there is MUCH which is wrong with it. – Did Jun 30 '14 at 22:00
  • 4
    It's only mildly confusing here but it gets extremely confusing once you start doing multiple integrals. What does $\displaystyle \int_0^x \int_0^x x^3 x^4 , dx , dx$ mean, for example? – Qiaochu Yuan Jun 30 '14 at 22:02
  • @QiaochuYuan You point to the real issue. In principle $\int_0^x(\int_0^xx^3dx)x^4dx$ is unambiguous. But mathematicians like the abuse of notation that allows them to write what you just wrote and they find it worthwhile to sacrifice $\alpha$-equivalence to get it. (See http://en.wikipedia.org/wiki/Lambda_calculus#Alpha_equivalence .) I suspect that modern mathematics students with some computing experience may have slightly different ideas of what's reasonable. – Dan Piponi Jun 30 '14 at 22:36

0 Answers0