The area bounded by the tangential lines to an even-power exponential function

Solution 1:

$y = x^m ~$ 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} $