I already know how to do this problem, but I have a question on a small part of the solution. This is an example out of Thomas' Calculus. I included the solution from the text below, and I'm wondering about the fact that the domain of $A(x)$ is $[0, 2].$ Wouldn't an $x$-value of $0$ or $2$ give a rectangle that is just a line, which wouldn't be a rectangle at all? Wouldn't this contradict the premise that we're looking for the largest "rectangle" that can be inscribed in a semicircle of radius $2?$ I feel like the domain should be $(0, 2).$ I know that this wouldn't change the answer at all, but it still bothers me, and it comes up all the time with these kinds of problems.
Best Answer
If you impose that a rectangle must have width and length greater than $0$, then your argument can be used to prove that there is no smallest rectangle in the given conditions. But since that restriction has no effect if we are searching for the largest rectangle, why bother?