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What is the arclength of the polar curve #f(theta) = sin(3theta)-4cot6theta # over #theta in [0,pi/4] #?

Answer 1

Cameron Alexander

The arclength is infinite.

From the reference Arc Length with Polar Coordinates

#L = int_alpha^beta sqrt{(r(theta))^2 + ((dr(theta))/(d theta))^2}d theta#

Given: #r(theta) = sin(3theta) - 4cot(6theta), alpha = 0 and beta = pi/4#

#(dr(theta))/(d"theta) = 3cos(3theta) + 24csc^2(6theta)#

Substituting into the integral:

#L = int_0^(pi/4) sqrt{(sin(3theta) - 4cot(6theta))^2 + (3cos(3theta) + 24csc^2(6theta))^2}d theta#

This integral does not converge.

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Answer 2

Christopher Allers

The arclength of the polar curve f(θ) = sin(3θ) - 4cot(6θ) over θ in [0, π/4] is approximately 0.774.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.