I'm a high school math teacher teaching an introductory calculus course, and I'm having a problem teaching one particular student about the physical definition of an integral.
The intuition is that it's the "area under the curve," and all but one of my students accept that this implicitly means "area under the curve down to the x-axis," but one student is hung up on thinking that "area under the curve" extends all the way down to $y = -\infty$.
I tried giving him the following proof:
Let $\int_0^1 f(x) \ dx$ be the area under $f(x)$ extending to $y = -\infty$. It is clear visually that $\int_0^1 f(x)-g(x) \ dx$ is the area between the functions $f(x)$ and $g(x)$. Now let $g(x) = 0$. We then have that $\int_0^1 f(x)-0 \ dx = \int_0^1 f(x) \ dx$ is the area under $f(x)$ extending to $y = 0$, contradicting our initial assumption.
He seems unconvinced by the above procedure. Is there any alternate phrasing I can use to convince him of this? I don't want him thinking that every integral evaluates to $\infty$ just because he's hung up on the wording.
I'm new to teaching and I've never had this misconception come up before, so I'm trying to fumble around with ways to properly explain this such that my other students don't doubt their correct intuition.