# Canonical "area between $f$ and $g$" question

This is a request of opinion from the Math.SE community about creating a new, canonical question that will answer all basic "find the area between $f$ and $g$" questions (for "simple," single-variable $f$ and $g$).

Purpose: To provide a single question, of which all elementary area problems presented in freshman calculus may be closed as a duplicate. These involve statements such as "find the area between $f(x) = e^x+10$ and $g(x) = \sin(x)$ for $x=0\ldots 3$."

To specify "elementary" problems: $f$ and $g$ should consist of easily integrable problems, and there shouldn't be a "trick" to evaluating an integral. (For example, "Find the area under $e^{-x^2}$ on $(-\infty, \infty)$" is out of the scope of this question.)
Problems that are certain to be in the scope of this question include those where $f$ and $g$ are linear combinations of $x^n$, $x^{n-1}e^{\alpha x^n}$, $x^ne^{\alpha x}$, $\sin x$, and $\cos x$. This is not exhaustive, but exemplifies what I'm talking about.

What I'm looking for: I'm asking this question first and foremost to gauge interest in this idea--does this sound like something the community wants?

I think this is a good idea for three reasons:

1. "Find area between __ and __" questions can literally be typed verbatim into Wolfram|Alpha. We don't need to be human computers and actually compute the answer to these questions! Rather, the asker really needs to learn how to solve this problem. If someone just wants the answer to cheat on their homework, they can type it in to Wolfram. If someone actually wants to learn, a canonical answer can teach how to solve the problem in a general way just as effectively as custom-written solutions.
2. As Math.SE receives more and more questions, we need to strategically choose what we answer. If we answer every question that's asked, the signal-to-noise ratio is unfavorable. A canonical answer will reduce a lot of "noise" questions that are very specific, and are most likely only helpful to other users as examples.
3. Most of the questions this affects are low-effort questions anyway. If you're in the camp that thinks low-effort/PSQs should be closed and destroyed, you can see why this helps your cause. If you like low-effort questions, see points 1 and 2 above. :)
• Potential candidates: "area between" questions with 1K+ views
– user147263
Jul 30, 2014 at 7:30
• As a side note: This also can apply to "area below" questions. (Granted, not many have 1K+ views, but it broadens the scope a bit.) (Possibly of interest to @900sit-upsaday, so I'll tag.) Jul 31, 2014 at 1:03
• And then we have two curves and an axis. The vertical axis could also come into play, bounding the area instead of intersection points. Or two vertical lines $x=a$, $x=b$. You see, when calculus textbooks devote an entire section to a topic, one Q&A pair can hardly suffice. The closest to canonical resource that I know is Paul's online notes, with seven carefully chosen examples. I think that a canonical Math.SE Q&A would probably be inferior to that.
– user147263
Jul 31, 2014 at 1:31
• Which is why I'm inclined to do the following: if there is a specific question (!= statement), answer that. Otherwise, VTC, leaving a link to Paul's Notes if one is feeling generous.
– user147263
Jul 31, 2014 at 1:31
• There is value in showing OP how to overcome a hurdle in a problem that OP has tried to solve. I believe that in most cases it is best to avoid carrying out the computations explicitly. Aug 1, 2014 at 4:30

For context, the list of newest questions containing "the area between".

I am not sure they can be reduced to a canonical one. The questions that contain nothing but problem statement are closed pretty quickly. Those that contain more than a problem statement usually request feedback on their approach. Answering "why is this wrong?" with "here's the right (generic) method" is somewhat unsatisfactory.

Also, even the problems with simple functions can be tricky. Locating the points of intersections is sometimes hard. And one may have to slice the region horizontally, but how to decide? And then curves cross each other in the middle, and all the hell breaks loose... These aspects are not so easy to catch at a glance. It's easy to recognize a homework dump in the close review queue, but figuring whether the problem indeed has no tricks in it (and thus is answered by the canonical example) pretty much requires solving it.

So, there are two factors limiting the usefulness of a canonical question here:

1. If the asker spent any prior effort on the problem (which includes opening the textbook and reading at least one canonical example) then their question probably has something to it that can't be answered by a canonical. Maybe the curves cross, or maybe they forgot that $\cos (0)=1$ and not $0$. Either way, some custom troubleshooting is called for.

2. If the asker spent no effort before asking the problem, then it can be closed as off-topic anyway.

Also, two technical points:

1. Questions closed as off-topic get automatically cleaned up more efficiently than those marked as a duplicate.

2. If such questions are closed as duplicates, any user on this list can single-handedly reopen and answer if they feel like it.

I upvoted this post (me?upvote?) because it's a good point to bring up, but I am leaning against implementing this idea.

• On the other hand, the "troubleshooting" type questions aren't of lasting value. I sometimes wish there was a graceful way for them to age away.
– user147263
Jul 30, 2014 at 6:06