Double Integral Bounds / Substitution

Question

I am having trouble following these steps in a reading on multivariable calculus.

Due to a change of variables:

$ \displaystyle\int_0^1 \int_0^s v^7 dv \, ds = \int_0^1 \int_s^1 s^7 dv \, ds$

Could anyone explain how you make the substitution to get to the right equation? What is the 'change of variable' that is applied? Thanks!!

What I tried

I've tried sketching a region, and applying substitutions, but what I'm stuck on is the order of integration isn't changing (dv ds both times), so I'm not sure how to rewrite it in this form.

One other note -- I realize you can evaluate each side independently and get the same answer, that's not what I'm worried about here. What I'm trying to understand is the change of variables that equates both sides of the equation.

Answer 1

As the letters used do not affect the result we can just swap the letter assigned to $v$ and $s$ $$\int_0^1 \int_0^s v^7 dv \, ds = \int_0^1 \int_0^v s^7 ds \, dv$$ and then by studying the regions of integration we have that $$0\le v\le1$$ $$0\le s\le v$$ So swapping the order of integration we get that $$s\le v \le 1$$ $$0\le s \le 1$$ and hence we get the required result $$\int_0^1 \int_0^s v^7 dv \, ds = \int_0^1 \int_0^v s^7 ds \, dv=\int_0^1 \int_s^1 s^7 dv \, ds$$

Answer 2

Assuming that my comment about describing the integral is correct. The integrals both involve integration over a triangle where the unit square cut in half by the diagonal $v=s$. The triangle is defined by $(s,v)=(0,0),(0,1),(1,1)$, using my definition of the right side integral $\int_0^1 \int_v^1v^7dsdv$.