Elliptic Integral of the Second Kind
Definition
The integral below is referred to as the complete elliptic integral of the second kind.
$$ E(k)=\int_{0}^{{\textstyle \frac{\pi}{2}}}\sqrt{1-k^{2} \sin ^{2} \theta} d\theta $$
The integral below is referred to as the incomplete elliptic integral of the second kind.
$$ E(\phi, k)=\int_{0}^{\phi}\sqrt{1-k^{2} \sin ^{2} \theta}d\theta $$
Explanation
The reason why the above two integrals are named elliptic integrals is that they emerge from the process of calculating the perimeter of an ellipse.
$$ \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1,\quad (0<a<b) $$
Given an ellipse,
$$ 4bE(k),\quad k^{2}=\frac{b^{2}-a^{2} }{b^{2}} $$
its perimeter can be calculated as shown. The graph below indicates the complete elliptic integral of the second kind as a function of $k$.
In the equation of an ellipse, if $a=b$, it becomes a circle since $E(0)=1.571$, hence the perimeter
$$ 4b\times 1.571=2\pi b $$
turns into the conventional formula for the circumference of a circle. Meanwhile, the incomplete elliptic integral represents the length of the arc of an ellipse up to a certain angle. However, the angle $\theta$ is different from the angle in conventional polar coordinates, as shown in the diagram below.
