# multi variable calculus, confused about changing limits of integration

I am really confused and would appreciate help on this....

I know if you have constants as limits of integration and you want to switch them you just switch the limits from dx to dy and switch the order of integration of variables.....

however i am really confused about whats going on below....how can you have functions as limits of integration and in this case what do you do to solve it because the below example doesnt make sense to me....

how are they finding new limit of integration because this seems if you have functions as limits you cant just switch them you have to do some weird stuff to get new limits, what do you do?

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The first integration on the RHS is holding $y$ constant. Therefore it is not a function in the limit, it is a "constant". Afterwards it is then an integration $wrt$ to $y$.