What is the area of lemniscate?

Find the area bounded by the lemniscate of Bernoulli r2 = a2 cos 2θ. The curve is symmetrical with respect to the origin, and occurs only with values of θ from -45° to 45° (-¼ π to ¼ π).

What is the equation of lemniscate?

Inverting the lemniscate in a circle centred at the origin and touching the lemniscate where it crosses the x-axis produces a rectangular hyperbola. The bipolar equation of the lemniscate is r r ′ = a 2 / 2 rr’ = a^{2}/2 rr′=a2/2.

How do you measure lemniscate?

Half of one lobe of the lemniscate is achieved when θ goes from 0 to π/4. Therefore the total length of the lemniscate of Bernoulli is given by: L=4∫π/40√r2+(drdθ)2dθ

How to calculate the area of Bernoulli’s lemniscate?

How do I calculate the area of Bernoulli’s Lemniscate? can anyone help me calculate this area? I have to use double integrals, and the question sounds like this: ” Calculate the area bounded by the curve ( x 2 + y 2) 2 = a 2 ( x 2 − y 2), where a is a real constant.

Is the Bernoulli equation constant at any point?

Bernoulli Equation. For a non-viscous, in-compressible fluid in a steady flow, the sum of pressure, potential and kinetic energies per unit volume is constant at any point. Bernoulli’s principle: At points along a horizontal streamline, higher pressure regions have lower fluid speed and lower pressure regions have higher fluid speed.

How to calculate the lemniscate of a hyperbola?

A lemniscate has the neat property that a normal (line at right angles) to the segment OP (where O is the origin and P is a point on the lemniscate) traces out the two arms of a hyperbola. x 2 − y 2 = 1. \\displaystyle {x}^ {2}- {y}^ {2}= {1}. x2 − y2 = 1.

How is the Bernoulli equation modified for the elevation?

If we assume that the gravitational body force is negligible – the elevation is small – then the Bernoulli equation can be modified to. p = p 1 + ρ v 1 2 / 2 = p 2 + ρ v 2 2 / 2 – p loss = p 1 + p d 1 = p 2 + p d2 – p loss (3) where . p = pressure (Pa, psi) p loss = pressure loss (Pa, psi) p d = 1/2 ρ v 2 = dynamic pressure (Pa, psi)