Recently I came upon a problem (if you would call it that, more of a thought experiment), which was phrased something like this:
Rotate the area formed by $\int_{-1}^12dx$ around the curve $h(x)=-x^{2}-1$.
First of all, I don't even know if this makes sense or is possible. But my friend and I thought that we could first "reflect" the given area "through" $h(x)$.
I am not sure if his method is correct. What it says is basically you take a point in the area given, then find its corresponding $h(x)$ value, and then find the distance between the two. After that you draw a normal vector to $h(x)$ at the x-value in question, where the magnitude of the vector is the distance between the two functions.
For example, take the point $(1,1)$ in the given area. First you would find $h(1)$, which equals $-2$, then do $1-(-2)=3$ to find the distance between the two points. After that, you draw a normal vector to $h(x)$ at $x=1$, where the magnitude of this vector is 3.
He calls this the "Adrian transformation" of a function (reflecting it "through" another function), and describes it as
$$-(f(t)-g(t))N(t)$$
where $f$ and $g$ are the functions you want to reflect and reflect through, respectively, and $N$ is the unit normal vector function to $f(t)$.
So my questions are:
- Does this "problem" make any sense at all?
- Does this method of normal vectors, etc. work?
- Is this just fancy for another, simpler, transformation?
I understand that this question is probably phrased poorly, please comment with questions and such so I can improve it. Thanks!!