Fundamental Theorem of Calculus Question (Area under Curve)

Question

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!

Answer 1

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.

Answer 2

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.