Hi I was looking at the soln. to this question and i got a little confused over the reversing of the integral. I would have thought that the reversing would come out to be 0<t<r and t<r<infinity, but that isn't the case. Could someone explain? Thanks
Here's what I think is going on....: We’re trying to evaluate an integral in the variable t. So when we get to the end of the solution we should get an answer independent of t (as we’ll have “integrated it out”). What’s wrong with the way you’ve chosen the limits is that if you had the outer integral running from t to infinity, then when you evaluated it by substituting in the limits you’d get something which would depend on t. So it would have gone wrong. In general, I think it’s the case that you should only have a variable on the limit for the inner integral, and let this guide the reversal of limits. The limits for the outer integral should be numbers. Hoping this makes sense!
The thing is I would have expected the reversal to go like the one on the top of page 38 in regards to the way the variables stay the same after the reversal on the integrals.
OK - I think I see now. The reversal of the limits on the integrals works in exactly the same way in 9.12 (from bottom of page 41 to top of page 42) as it does in 9.8 (from bottom of page 37 to top of page 38). BUT!! In 9.12 at the same time as reversing the integrals they also swap the variables r and t over. I can't really see why this is necessary - it's as if here they always want the variable in the inner integral to be r and in the outer integral to be t. Not swapping r and t when reversing the integrals doesn't change the answer, though. *Ted*
