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I was in my Grade $12$ Calculus class today and we were learning about how to find the area under a curve. It included a lot of the questions of the type, "Find the area of the curve $f(x)$ from $x = ...$ to $x = ...$, bounded by the $x$-axis" and "Evaluate the following definite integral using the Fundamental Theorem of Calculus".

My teacher then said, "For your homework questions, don't always assume that the curve meets the $x$-axis." What does he mean by this? I don't really understand.

Thank you!

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    $\begingroup$ Did you ask your teacher? $\endgroup$ – mathguy Jan 20 '17 at 0:13
  • $\begingroup$ I did not but he is currently away so I have to resort to Stack Exchange for now. $\endgroup$ – user164403 Jan 20 '17 at 0:14
  • $\begingroup$ @user164403 Do you see it now? $\endgroup$ – TripleA Jan 20 '17 at 0:19
  • $\begingroup$ I don't personally know why your teacher would warn you of this. You still need to check if the function touches the x-axis, as you would if your teacher didn't warn you. Are you sure you understood correctly? $\endgroup$ – Kaynex Jan 20 '17 at 1:01
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I think your teaching is implying something like $x^2+1$ from $x=0$ to $x=1.$ It's a parabola that has been shifted up by $1$.

He may also be trying to get you prepared for "nastier" looking things like the following here.

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    $\begingroup$ If I was in the OP's shoes, my question would be "Why warn me not to assume the graph meets the $x$-axis? Why would I ever assume that in the first place?" I would be just as confused by that advice as the OP is. $\endgroup$ – mathguy Jan 20 '17 at 0:19
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enter image description here

As you can see here, the curve never touches the axis but using basic integration, you still can find the area between $a$ and $b$

Complications arise when the curve touches the x-axis and it's then the negative integral. So you have to take into consideration which areas are above or below the $x$ axis.

enter image description here

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  • $\begingroup$ Of course, there are curves where they meet neither axis, but you'll get to that eventually. $\endgroup$ – TripleA Jan 20 '17 at 0:17
  • $\begingroup$ Thank you for your explanation! I see what you mean, but I don't know why my teacher said this (maybe I'm not seeing any potential complications or mistakes that a student would make while solving these questions...). $\endgroup$ – user164403 Jan 20 '17 at 0:22
  • $\begingroup$ @user164403 Hopefully, the edit will be useful to you then $\endgroup$ – TripleA Jan 20 '17 at 0:25
  • $\begingroup$ Oh, so basically how I understand it is the complication would be that if the curve is going above the x-axis, and then suddenly intersects the x-axis and has area under the curve, the area of the curve would be the positive area minus the negative area when in reality area is an absolute value? Thank you for your effort. $\endgroup$ – user164403 Jan 20 '17 at 0:26
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    $\begingroup$ @user164403 That's exactly correct, well done. $\endgroup$ – TripleA Jan 20 '17 at 0:28

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