Suspicious Dark Spot

There is some water in a tank, and on the surface of the water, there is a ripple. (Depicted in the above gif)

The ripple can be described with the Cartesian coordinate system as \[z=p\left(\sqrt{x^2+y^2}\right) \qquad \text{ where } \qquad p\left(x\right)=-\frac{5\sin \left(x\right)}{x}+10 \; .\]

The bottom of the tank is described as \(z=0 \).

Now, rays of light parallel to the \(z\)-axis shine onto the surface of the water. The rays of light are then refracted and travel through the water, after that hitting the bottom of the tank, this forms a circular dark spot of light with its center at coordinates \((0,0,0)\).

Find the area of the dark spot, give your answer to 3 significant figures.

Details and Assumptions:

  • The refractive index of water is 1.33.

  • The dark spot is bounded by a bright circle around its perimeter.

  • You may use a computational method.

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