(I found this similar question unhelpful.)
Here is a sketch of a situation that pertains to my question:
- $f\left( x\right)$ is a parabola of the form $y = x^2 - 4x + 6 = \left( x-2 \right) ^2 +2$ in the $xy$-plane for $x \in \left[ 0,4 \right]$
- $f\left( x \right)$ has two endpoints: $\left( 0,0 \right)$ and $\left( a, f\left( a \right) \right) = \left( 4,6 \right)$
- The third point of significance on $f \left( x \right)$ is $\left( b, f \left( b \right) \right) = \left( \sqrt 6, 12-4\sqrt 6 \right)$, where the line tangent to $f \left( x \right)$ that also intercepts the origin has smallest slope
- $\alpha$ is the angle between the $y$-axis and the line that connects the origin and $\left( a, f\left( a \right) \right)$
- $\beta$ is the angle between the $y$-axis and the line that connects the origin and $\left( b, f\left( b \right) \right)$
My question is which angle ($\alpha$, $\beta$, or another) is the angle that $f\left( x \right)$ subtends with respect to the origin, and why?
Also, if possible, could you use this as an example of how to calculate subtended plane angle using the formula
$$\theta = \int \frac{\hat r \cdot \hat n}{r} \, d\ell$$
as outlined by the sketch below:
