I have a problem with the integral $\int_0^1 \int_0^{t^2}x^2 t^2 dx dt .$The problem I have with this integral is that in the description of the exercise it said that we should evaluate this (I did that) and after that we should change the order of integration - it also said that we should beware of the limits.
MY question is now: What exactly should I change when integrating ? Should I integrate
i) $\int_0^1 \int_0^{t^2}x^2 t^2 dt dx $ or
ii) $ \int_0^{t^2} \int_0^1 x^2 t^2 dt dx $ or
iii) $ \int_0^{t^2} \int_0^1 x^2 t^2 dx dt $
(all possible combinations of changing the integral and the $dx,dt$'s) ? And how does this relate to my first integral ?
We did a theorem about changing the order of integration, but only if the domain of integration is fixed (whereas here it depends on $t$) - if I could apply that theorem, then iii) would be the correct integral that is equal to the first one.
Please tell me what the scope of this exercise was, since our professor didn't give us any explanation.