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Calculus

Integrals

Since the area of a curve in polar coordinates ρ(θ) between the angles α and β is

[11] A = 12 αβρ2dθ

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

[12] A =n22θ+sin(4θ)+2sin(2θ)-23sin3(2θ) α β

Total area

The total area of the siluroid is 2πn2

Main lobe

The area of the main lobe is

[13] π + 83n2

Secondary lobe

The area of a secondary lobe is

[14] 12π -  83n2

Derivatives

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

[15] ρ˙ = -4nsinθ3cos2θ + 2

whereas the second derivative is

[16] ρ¨ = -4ncosθ9cos2θ - 4

First derivative

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

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Second derivative

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

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