How do you find 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.

How do you find the limit of a polar area?

The area of a region in polar coordinates defined by the equation r=f(θ) with α≤θ≤β is given by the integral A=12∫βα[f(θ)]2dθ. To find the area between two curves in the polar coordinate system, first find the points of intersection, then subtract the corresponding areas.

How to calculate arc length in polar coordinates?

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 for is given by In polar coordinates we define the curve by the equation where In order to adapt the arc length formula for a polar curve, we use the equations

How to find the arc length of a curve?

In this section we’ll look at the arc length of the curve given by, where we also assume that the curve is traced out exactly once. Just as we did with the tangent lines in polar coordinates we’ll first write the curve in terms of a set of parametric equations, and we can now use the parametric formula for finding the arc length.

How to calculate the area under a polar curve?

Now we turn our attention to deriving a formula for the area of a region bounded by a polar curve. Recall that the proof of the Fundamental Theorem of Calculus used the concept of a Riemann sum to approximate the area under a curve by using rectangles.

Which is the integral for the arc length?

The integral for the arc length is then, L = ∫ 1 0 √ 1 + y 2 d y. This integral will require the following trig substitution. y = tan θ d y = sec 2 θ d θ y = 0 ⇒ 0 = tan θ ⇒ θ = 0 y = 1 ⇒ 1 = tan θ ⇒ θ = π 4 √ 1 + y 2 = √ 1 + tan 2 θ = √ sec 2 θ = | sec θ | = sec θ.