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The arc length of polar functions: Find the arc lengths of the following polar functions.

r=secฮธforฮธโˆˆโˆ’ฯ€6,ฯ€6

Short Answer

Expert verified

The arc length is23units

Step by step solution

01

Given information.

The given curve isr=secฮธforฮธโˆˆโˆ’ฯ€6,ฯ€6

02

Arc length

The arc length of the curve is given by,

L=โˆซฮฑฮฒr2+drdฮธ2dฮธL=โˆซ-ฯ€6ฯ€6secฮธ2+secฮธtanฮธ2dฮธโˆตdrdฮธ=secฮธtanฮธL=โˆซ-ฯ€6ฯ€6sec2ฮธ+sec2ฮธtan2ฮธdฮธL=โˆซ-ฯ€6ฯ€6sec2ฮธ1+tan2ฮธdฮธsin2ฮธ+cos2ฮธ=1L=โˆซ-ฯ€6ฯ€6sec2ฮธsec2ฮธdฮธ1+tan2ฮธ=sec2ฮธL=โˆซ-ฯ€6ฯ€6sec2ฮธdฮธL=tanฮธ-ฯ€6ฯ€6L=tanฯ€6-tan-ฯ€6L=tanฯ€6-ฯ€-tanฯ€6L=2tanฯ€6-ฯ€L=23units

The arc length isL=23units

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