I don't have issues with doing integration problems, but occasionally I see the solution changing the limits of integration whenever a $u$-substitution is done.
I obviously don't have a problem doing this, and I just recently noticed my book doing this under the chapter involving "area of surface of revolution."
My question is did I develop a bad habit by never changing the limits of integration, or is it a best practice to always change limits of integration?
My teacher said on a test, and in general, if we do not change the limits of integration then we should be signifying this by labeling our limits of integration $x=$ lower-limit and $x=$upper-limit.
EDIT EXAMPLE INCLUDED
After further investigation, my confusion is because of the two below equations:
Find the exact area of the surface obtained by rotating the curve about the x-axis
$$y=\sqrt{1+4x}, 1\le x\le 5$$
The limits of integration were changed in the solution to this problem.
The given curve is rotated about the y-axis. Find the area of the resulting surface.
$$y=x^\frac{1}{3}, 1\le y \le 2$$
The limits of integration were NOT changed in the solution to this problem.
Best Answer
Sometimes we are doing an indefinite integral by $u$-substitution, and then the form of integral involving $u$ is easily converted back into $x$ or whatever the original variable was at the end of the process.
However when a definite integral is involved, you have a choice of either converting the limits of integration from (say) $x$ limits to $u$ limits, or considering the $u$-substitution as a means to obtaining the final indefinite integral in terms of $x$ and using the original limits of integration.
The former has the advantage of skipping the substitution back into $x$, but at the cost of figuring out how to change the limits of integration into terms of $u$. This was a bad habit you learned, or more precisely, a useful habit you failed to learn. Converting the limits from $x$ to $u$ is ordinarily just a matter of using the $x$ limits in the expression for $u$ in terms of $x$.