In this lesson, we will learn how to find the length of a curve defined by polar equations using integration.
Students will be able to
Q1:
Write the integral for the arc length of the spiral 𝑟=𝜃 between 𝜃=0 and 𝜃=𝜋. Do not evaluate the integral.
Q2:
The purpose of this question is to get improved estimates on the length of a spiral curve.
Use the fact that 𝑥<1+𝑥<(1+𝑥) when 𝑥>0 to find lower and upper bounds for the length 𝐿 of the spiral 𝑟=𝜃 between 𝜃=0 and 𝜃=𝜋. Give your answer to 4 decimal places.
By comparing √1+𝑥 to the average of 𝑥 and 1+𝑥 when 𝑥>0, find better bounds for estimating 𝐿. Give your answer to 4 decimal places.
Q3:
Find the arc length of the polar curve 𝑟=𝜃+𝜃sincos, where 𝜃 lies in the interval [0,𝜋].
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