# How to apply change of variables to identity map?

I'm confused about how to apply the change of variables theorem to calculating areas using the identity function. For instance, if $T$ is a linear map (Jacobian is constant) then I think change of variables says $$\int_{T(A)}1=|\det V|\int _A 1\circ T.$$ But then $1\circ T=T$ and the RHS has $\int _AT$ which doesn't seem right... Should $1$ be replaced by an indicator? What's going on?

We have $$1\circ T = 1$$ not $1 \circ T = T$. $1$ is not the identity it is a constant function of value $1$.