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Fill in the blanks to complete each of the following theorem statements:

Let r=fθbe a differentiable function of θsuch that f'(θ)is continuous for all θα,βFurthermore, assume that r=f(θ)is a one-to-one function from α,βto the graph of the function. Then the length of the polar graph of r=f(θ)on the interval α,βis..............

Short Answer

Expert verified

The blank is filled byαβf'θ2+fθ2dθ

Step by step solution

01

Step 1. Given information

r=fθ

αθβ

02

Step 2. Write formula of length of the arc.

As we know that when x and y are functions of the parameter θ, the arc length of the curve on the interval [α, β] is

l=αβdxdθ2+dydθ2dθ

By combining this integral with the parametric equations for the curve,

x=rcosθx=fθcosθand y=fθsinθ

we have an arc length

l=αβddθf(θcosθ)2+ddθf(θsinθ)2dθ=αβ-f(θ)sinθ+f'(θ)cosθ2+f'(θ)sinθ+f(θ)cosθ2dθ

After expanding the terms under the radical and simplifying , we get:

l=αβf'θ2+fθ2dθ

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