Let the area of a region ‘R’ bounded by the curves in polar coordinatesystem beRrdrd . Find the area of one loop of the lemniscate (as shown infigure)2

The lemniscate is given by the polar equation r^2 = a^2 cos(2θ). The area of one loop of the lemniscate can be found by integrating the area element in polar coordinates, 1/2 r^2 dθ, over the region of the loop.

Step 1: Identify the limits of integration. One loop of the lemniscate is traced out as θ goes from -π/4 to π/4.

Step 2: Set up the integral. The area A of one loop is given by the integral

A = ∫ (from -π/4 to π/4) 1/2 r^2 dθ = ∫ (from -π/4 to π/4) 1/2 (a^2 cos(2θ)) dθ

Step 3: Evaluate the integral. Using the power-reduction identity, cos(2θ) = 2cos^2(θ) - 1, we can rewrite the integral as

A = a^2/2 ∫ (from -π/4 to π/4) (cos^2(θ) - 1/2) dθ = a^2/2 [θ/2 - sin(2θ)/4 - θ/2] (from -π/4 to π/4) = a^2/2 [π/4 - 0 - (-π/4 - 0)] = a^2/2 * π/2 = a^2π/4

So the area of one loop of the lemniscate is a^2π/4.