For $$I_A=\int_a^bf(x)dx\int^b_ag(y)dy$$and$$ I_B=\int^b_af(x)g(x)dx$$
I'm trying to work out what is the relationship between these two integrals. What I'm thinking is considering $I_B$ as the area under the $I_A$'s diagonal line, which is $y=x$. However, that would be wrong since the length of the diagonal line is $\sqrt 2(b-a)$ instead of $b-a$. So is there any intuitive relationship between this two integrals?
Pick $c\in\mathbb{R}$. Define $f'\left(x\right)=cf\left(cx\right)$ and $g'\left(x\right)=g\left(cx\right)$ (prime not a derivative) so our new integrals are $$I_{A}'=\int_{ca}^{cb}cf\left(cx\right)dx\int_{ca}^{cb}g\left(cx\right)dx$$ and $$I_{B}'=\int_{ca}^{cb}cf\left(cx\right)g\left(cx\right)dx$$ We can see that $I_{A}'=\frac{1}{c}I_{A}$ and $I_{B}'=I_{B}$. So, we can choose $I_{A}$ arbitrarily and hold $I_{B}$ fixed. In other words, there is no relationship.