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Calculus

Integrals

Since the area of a curve in polar coordinates $\rho \left(\theta \right)$ between the angles $\alpha$ and $\beta$ is

[11]

it is possible to demonstrate that the area of the siluroid between two generic angles is

[12]

Total area

The total area of the siluroid is $2\pi {n}^{2}$

Main lobe

The area of the main lobe is

[13]

Secondary lobe

The area of a secondary lobe is

[14]

Derivatives

Let us use the equation in polar coordinates to calculate the derivatives of a siluroid. The first derivative is

[15]

whereas the second derivative is

[16]

First derivative

The curve corresponding to the first derivative of the mother equation is

Second derivative

The curve corresponding to the second derivative of the mother equation is

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The Siluroid

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