What is the arc length of a polar curve?
The arc length of a polar curve r=f(θ) between θ=a and θ=b is given by the integral L=∫ba√r2+(drdθ)2dθ. In the following video, we derive this formula and use it to compute the arc length of a cardioid.
What does a polar curve look like?
Polar curves are defined by points that are a variable distance from the origin (the pole) depending on the angle measured off the positive x-axis. Polar curves can describe familiar Cartesian shapes such as ellipses as well as some unfamiliar shapes such as cardioids and lemniscates.
Is arc length the same as length of curve?
We usually measure length with a straight line, but curves have length too. A familiar example is the circumference of a circle, which has length 2 π r 2\pi r 2πr for radius r. In general, the length of a curve is called the arc length.
What are the types of polar curves?
There are five classic polar curves: cardioids, limaҫons, lemniscates, rose curves, and Archimedes’ spirals.
What are the different types of arc length?
Arcs can be major, semicircular, or minor. Every arc corresponds to a central angle (angle whose vertex is the center of the circle).
How to calculate the arc length of a curve?
Here we derive a formula for the arc length of a curve defined in polar coordinates. In rectangular coordinates, the arc length of a parameterized curve (x(t), y(t)) for a ≤ t ≤ b is given by L = ∫b a√(dx dt)2 + (dy dt)2dt. In polar coordinates we define the curve by the equation r = f(θ), where α ≤ θ ≤ β.
How to calculate the length of a polar curve?
The key to computing the length of a polar curve is to think of it as a parametrized curve with parameter θ. (When computing the slope of a polar curve, we called the parameter t and set θ = t. Calling the parameter θ is equivalent and saves a step.)
How to calculate the arc length of a cardioid?
L = ∫ a b r 2 + ( d r d θ) 2 d θ. In the following video, we derive this formula and use it to compute the arc length of a cardioid. If playback doesn’t begin shortly, try restarting your device.
How to calculate the area between the curves?
To calculate the area between the curves, start with the area inside the circle between θ = π 6 and θ = 5π 6, then subtract the area inside the cardioid between θ = π 6 and θ = 5π 6: