What is distance around the graph of r = 3 sin θ
a) 3π
b) 9π
c) 12π
d) 72π
It is a circle tangent to the theta=0 axis, of diameter r=1.5
arc length= PI*d=3pi
now to integrate it use this example, just change the limits to zero to PI. https://socratic.org/questions/how-do-you-find-the-exact-length-of-the-polar-curve-r-3sin-theta-on-the-interval
The distance around the graph of r = 3 sin θ can be found by calculating the arc length of one complete revolution.
A complete revolution in polar coordinates covers an angle of 2π radians (or 360 degrees).
To find the arc length, we can use the formula:
Arc length = ∫√(r² + (dr/dθ)²) dθ,
where r = 3 sin θ.
First, let's find dr/dθ by taking the derivative of r with respect to θ:
dr/dθ = 3 cos θ.
Now, we can substitute the values into the arc length formula:
Arc length = ∫√(3 sin θ)² + (3 cos θ)² dθ
= ∫√(9 (sin² θ + cos² θ)) dθ
= ∫√9 dθ
= ∫3 dθ
= 3θ.
Since we are taking one complete revolution, θ ranges from 0 to 2π.
Therefore, the distance around the graph is:
Distance = 3 * (2π - 0)
= 6π.
The correct answer is not listed among the options provided.
To find the distance around the graph of the equation r = 3 sin θ, we need to integrate the arc length formula from θ = 0 to θ = 2π.
The arc length formula in polar coordinates is given by:
L = ∫[a, b] √(r^2 + (dr/dθ)^2) dθ
where a and b are the starting and ending angles, r is the function that describes the distance from the origin, and dr/dθ is the derivative of r with respect to θ.
In this case, our starting angle is 0 and our ending angle is 2π. The function r is given as r = 3 sin θ.
First, let's find the derivative of r with respect to θ:
dr/dθ = 3 cos θ
Now we can substitute r and dr/dθ into the arc length formula:
L = ∫[0, 2π] √(r^2 + (dr/dθ)^2) dθ
L = ∫[0, 2π] √((3 sin θ)^2 + (3 cos θ)^2) dθ
L = ∫[0, 2π] √(9 sin^2 θ + 9 cos^2 θ) dθ
L = ∫[0, 2π] √(9(sin^2 θ + cos^2 θ)) dθ
L = ∫[0, 2π] √(9) dθ
L = 3√(∫[0, 2π] 1) dθ
L = 3(θ)∣[0, 2π]
L = 3(2π - 0)
L = 6π
Therefore, the distance around the graph of r = 3 sin θ is 6π, which is not among the given options.
come on - it's just a circle of radius 3/2
You should be able to recognize that by now...