Lesson: Arc Length of a Polar Curve

Mathematics

In this lesson, we will learn how to find the length of a curve defined by polar equations using integration.

Worksheet: 25 Questions

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:

Let 𝑠 be the arc length of the polar curve 𝑟=3𝜃 over the interval 0≤𝜃≤𝜋2. Express 𝑠 as a definite integral.

Using a calculator, or otherwise, find the value of 𝑠 giving your answer to 4 decimal places.
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