Change variable of integration from dy to d(y/δ)

[Saladsamurai]

Homework Statement

I have this mess of an equation:

$$\frac{d\delta}{dx}\int_{y=0}^\delta g\,g'\frac{y}{\delta^2}\,dy + \frac{d\delta}{dx}\int_{y=0}^\delta g'\frac{y}{\delta^2}\,dy = g'\frac{1}{\delta}|_{y=0}^\delta \qquad(1)$$

and I want to change the variable of integration from y to y/δ. Also note that g is a function of the independent variable (y/δ)

Homework Equations

I know that my limits must change from 0 to y into 0 to 1.

The Attempt at a Solution

$$\frac{d\delta}{dx}\int_{y/\delta=0}^1 g\,g'\frac{y}{\delta^2}\,d(y/\delta) + \frac{d\delta}{dx}\int_{y/\delta=0}^1 g'\frac{y}{\delta^2}\,d(y/\delta) = g'\frac{1}{\delta}|_{y/\delta=0}^1 \qquad(2)$$Note that in (2), all I did was change the limits and I replaced dy everywhere with d(y/δ).

Is that correct? I feel like I am missing something else here.

 

[cepheid]

I think this is how the change of variables should go:

Let σ(y) = y/δ

dσ = d(y/δ) = dy/δ (edit: have assumed that δ is indep. of y here)

→ dy = δdσ = δd(y/δ)

So, everywhere in your original equation that you see dy, you must replace it with δd(y/δ). If I understand you right, you have not included these factors of δ in front.

Now for the limits of integration:

lower limit: σ(y=0) = 0/δ = 0

upper limit: σ(y=δ) = δ/δ = 1

So you did those right.

 

[Saladsamurai]

Nice catch cepheid! I had tried doing something similar but somehow worked myself in a circle!

Thanks