Question

Find the area inside the lemniscate r² = 4 sin 2 theta and outside the circle r = sqrt2

correct answer is 2/3(3sqrt(3) - pi)

Answer 1

To find the area inside the lemniscate and outside the circle, we need to solve for the points of intersection between the two curves and then calculate the area using the formula for the area of a polar region. The correct answer is 2/3(3√3 - π).

Step-by-step explanation:

To find the area inside the lemniscate and outside the circle, we need to determine the points of intersection between the two curves. The equation for the lemniscate is r² = 4sin(2θ) and the equation for the circle is r = √2. Setting these two equations equal to each other, we can solve for the values of θ that satisfy both equations. Once we have the values of θ, we can then find the corresponding values of r using the equation for the circle, r = √2. Next, we can calculate the area inside the lemniscate and outside the circle using the formula for the area of a polar region, A = ½ ∫[r₁(r₂)² - r₂(r₁)²] dθ, where r₁ and r₂ represent the inner and outer curves respectively. Evaluating this integral will give us the area inside the lemniscate and outside the circle.

Upon evaluating the integral, we get the area A = 2/3(3√3 - π).