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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?

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

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    $\begingroup$ I should have noticed that :'( $\endgroup$ – Calc Feb 5 '17 at 21:23
  • $\begingroup$ @Calc Honestly, this was the reason that I knew what you did wrong. It happens to all of us. But important is, that we investigate and so we will always remember if the same sitation occurs again. This is entirely normal. $\endgroup$ – TheGeekGreek Feb 5 '17 at 22:19

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