Use of Newton method to find the value of $x$ A segment of a circle is the region enclosed by an arc and its chord (See figure below). If $r$ is the radius of the circle and $x$ the angle subtended at the center of the circle, find the value of $x$ (correct to $4$ decimal places) for which the area of the segment is one-fourth the area of the circle. Use the Newton iteration method with $x_0 = 2$.

 A: Hint: To apply Newton's Method, we first need an equation for the area of the segment:
Area of the sector of the circle between the radii: $\frac12xr^2$
Area of the triangle inside the sector outside the segment: $\frac12r^2\sin(x)$
Area of the segment: $\frac12r^2(x-\sin(x))$
Area of one-fourth of the circle: $\frac14\pi r^2$  
So the equation we need to solve is $x-\sin(x)=\frac12\pi$ .
Apply Newton's Method to this equation.
A: Given $x$, we can find the area of the section as
$$
A = \frac{1}{2}r^2(x-\sin x)
$$
So, given $A$, the equation we'd like to solve (using Newton's method) is
$$
\frac{1}{2}r^2(x-\sin x)-A=0
$$
Or, as I will rewrite for convenience:
$$
x-\sin x-\frac{2A}{r^2}=0
$$
Now, if $A$ is one quarter the area of the circle, the above becomes
$$
x-\sin x-\frac{2(\frac{\pi r^2}{4})}{r^2}=0\implies\\
y(x) = x-\sin x-\frac{\pi}{2}=0
$$
Newton's algorithm dictates that we update $x$ by
$$
x_{i+1} = x_i-\frac{y(x_i)}{y'(x_i)}
$$
$y$ is as given above, so that $y'$ is given by
$$
y'(x) = 1-\cos x
$$
So, our method can be described recursively as
$$
x_{i+1} = x_i-\frac{x_i-\sin x_i-\frac{\pi}{2}}{1-\cos x_i}
$$
With the initial estimate $x_0=2$.
