For the curve r = 2 sin 3(theta):
Find the arc length of one petal
To find the arc length of one petal of the curve r = 2 sin(3θ), we can use the arc length formula for polar curves.
The arc length formula for polar curves is given by:
L = ∫√(r^2 + (dr/dθ)^2) dθ
In this formula, r represents the function that defines the polar curve, and dr/dθ represents its derivative with respect to θ.
Let's start by finding the derivative of r with respect to θ, which is dr/dθ:
Given r = 2 sin(3θ), we calculate dr/dθ by applying the chain rule.
dr/dθ = (d/dθ)(2 sin(3θ))
dr/dθ = 6 cos(3θ)
Now we substitute the values of dr/dθ and r into the arc length formula to find the arc length of one petal:
L = ∫√(r^2 + (dr/dθ)^2) dθ
L = ∫√(2 sin(3θ)^2 + (6 cos(3θ))^2) dθ
Since we are looking for the arc length of one petal, we need to integrate over one complete petal. In this case, we can integrate from θ = 0 to θ = π/6. This will cover one complete petal for the given equation.
L = ∫[0 to π/6] √(2 sin(3θ)^2 + (6 cos(3θ))^2) dθ
Now, you can evaluate the above integral using algebraic manipulation and trigonometric identities in order to find the arc length of one petal for the curve r = 2 sin(3θ).