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What is the formula for computing the arc length of a polar curve r=f(θ)where θα,βWhat conditions on the polar functionf(θ)are necessary for this formula to hold?

Short Answer

Expert verified

The formula for the arc length of the polar graph in the interval α,βis:

Arc length=αβf'(θ)2+(f(θ))2dθ

The required conditions for the polar function f(θ)are necessary for this formula to hold:aβf'(θ)2+(f(θ))2dθ

Step by step solution

01

Find the formula for the arc length of the polar graph in the interval[α,β] 

The arc length is the distance measured along the curved line that makes up the arc.

A polar curve's arc length is defined as follows.

Let r=f(θ)is any differentiable function of θsuch that f'(θ)is continuous for all θ[α,β]

Moreover, consider that r=fis a one-one function from [α,β]to the function graph. then in the interval [α,β]the arc length of the polar graph,

Arc length=αβf'(θ)2+(f(θ))2dθ

02

The objective is to find What conditions on the polar function r=f(θ) are necessary for this formula to hold?

To determine the length of the arc for the curve r=f(θ)the function f(θ)must be differentiable and one to one.

The derivative f(θ)of that is f'(θ)is continuous for all θ[α,β].

As a result, for this formula to work, f(θ)must be differentiable, continuous, and one to one.

Therefore the answer is:

aβf'(θ)2+(f(θ))2dθ

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