Question

Find the area enclosed by one leaf of the rose r=3cos(3θ).

a) 3 square units
b) 6 square units
c) 9 square units
d) None of the above

Answer 1

To find the area enclosed by one leaf of the rose r=3cos(3θ), we need to calculate the definite integral of half the square of the radius from θ = 0 to θ = π/6.

Step-by-step explanation:

The equation r = 3cos(3θ) represents the polar coordinates of a rose curve. To find the area enclosed by one leaf of the rose, we need to calculate the definite integral of half the square of the radius from θ = 0 to θ = π/6, since the curve is symmetric over this interval.

The definite integral is given by: Area = (1/2)∫[0,π/6] (r^2)dθ.

By substituting the equation for r, we get: Area = (1/2)∫[0,π/6] (9cos^2(3θ)dθ.

Solving this integral will give us the area enclosed by one leaf of the rose curve.