Question

Find the area of the region inside the lemniscate r²=6cos2θ and outside the circle r= 3 Sketch a graph of the region bounded by the graphs of the equations. Use: (i). Single integral in polar coordinate system. [Verify using Mathematica] (ii). Double integral in polar coordinate system. [Verify using Mathematica]

Answer 1

To find the area of the region inside the lemniscate and outside the circle, you can use both a single integral and a double integral in the polar coordinate system.

Step-by-step explanation:

To find the area of the region inside the lemniscate and outside the circle, we can use both a single integral and a double integral in the polar coordinate system. I'll explain both methods:

Single Integral Method:

First, we need to find the limits of integration. The lemniscate intersects the circle at two points, so we can use those points to determine the limits. We solve the equations r²=6cos2θ and r=3 simultaneously to find the values of θ where they intersect. Next, we integrate the equation 6cos2θ - 3² with respect to θ, and multiply the result by 0.5 to get the area of the region inside the lemniscate.

Double Integral Method:

Using the double integral method, we set up the integral by finding the limits of integration for r and θ. The limits for r are from 0 to 3 (the radius of the circle), and for θ, we use the values we found in the single integral method. Then, we integrate the equation r dr dθ with these limits to find the area of the region inside the lemniscate.