Question

​​​​​​​Let r(θ) = cos( 3 2 θ). The figure shows the planar curve parametrized by [0, 4π] → R2 , θ 7→ (r(θ) cos θ, r(θ) sin θ). Determine the area of the white, non-shaded parts of the ‘petals

Answer 1

To determine the area of the white, non-shaded parts of the petals of the curve, we need to find the area enclosed by each petal and subtract the area of the shaded parts. Using the given parametric equations, we can integrate to find the area of one petal and then multiply by 4 to get the total area of all four petals.

Step-by-step explanation:

To determine the area of the white, non-shaded parts of the petals of the curve, we need to find the area enclosed by each petal and subtract the area of the shaded parts. Since the curve is given by the parametric equations x = r(θ)cos(θ) and y = r(θ)sin(θ), we can rewrite the parametric equations in terms of θ and calculate the area using integration.

The curve r(θ) = cos(3/2θ) represents a rose curve with 4 petals. To find the area of one petal, we can integrate the equation A = 1/2 ∫(r(θ))^2 dθ from 0 to π/2, and then multiply by 4 to get the total area of all four petals.

Using the given equation for r(θ), we can square it and integrate from 0 to π/2:
A = 2∫[cos(3/2θ)]^2 dθ
A = 2∫cos^2(3/2θ)dθ
A = 2∫(1/2 + 1/2cos(3θ))dθ
A = ∫(1 + cos(3θ))dθ
A = θ + (1/3)sin(3θ)

Next, we calculate the area for θ = π/2:
A(π/2) = (π/2) + (1/3)sin(3(π/2))

To calculate the total area of all four petals, we multiply the area of one petal by 4:
A_total = 4(A(π/2))