The problem:
Consider a function, $f(x) = x^m$, where $m$ is an even, natural number. Then, consider two tangential lines, grazing the points $(-x,y)$ and $(x,y)$, respectively. My question is, what is the area bounded by those tangential lines and the curve, for any given $m,x$:
The area in-question has been colored yellow in the picture above. In this example, $m = 2$, $x = 1.5$ and $y = 9$.
My current research:
The function for a tangential line is denoted as $t(x)$. The $y$-value of the tangential point between $t(x)$ and $f(x) = x^2$ is denoted as $p_t$.
$$t(x) = -p_t + 2x\sqrt{p_t}$$
$p_t$ is met when $\displaystyle x = \sqrt{p_t}$:
$$-p_t + 2p_t = p_t$$
When I looked at this, I thought I could generalize it to any even $m$. That generalization looks like this:
$$t(x) = -p_t + 2x(\root m\of{p_t})^{m-1}$$
So, when $x = \root m\of{p_t}$, then $t(x) = p_t = x^m$. Problem is, when I tried this with $m = 4$ and $x = 3$, it created a secant line, not a tangent line. Not sure why. It did cross the curve at $(3, 81), 81 = p_t$, but it also crossed it much further below, somewhere in the interval $(7,7.2)$. So, that's where I'm at, trying to figure out a generalized formula for the function of any tangential line to an exponential function of the form $f(x) = x^{2n}$.
And that's simply my current sub-question, within the larger question of what is this area?
$y = x^m ~$ where $m \in \mathbb{N}$ and $m$ is even.
The curve has symmetry about y-axis so we can find the bound area to the right of y-axis and multiply by $2$.
Slope of the curve is $~y' = mx^{m-1}$
Equation of tangent at point $P ~(p, p^m), ~p \gt 0~$:
$(y - p^m) = m p^{m-1} (x-p)$
$\implies y = mp^{m-1} x - (m-1)p^m$
Height of shell bound between tangent line and the curve is,
$h(x) = x^m - mp^{m-1} x + (m-1)p^m$
So the bound area between the curve and the tangents at points $(p, p^m)$ and $(-p, p^m)$ is given by,
$A = \displaystyle 2 \int_0^p h(x) ~dx = \frac{m (m-1) p^{m+1}}{m+1} $