In this lesson, we will learn how to use the polar arc length formula for a parametric curve to find its length.
Q1:
Write the integral for the arc length of the spiral π = π between π = 0 and π = 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 π₯ < 1 + π₯ < ( 1 + π₯ ) 2 2 2 when π₯ > 0 to find lower and upper bounds for the length πΏ of the spiral π = π between π = 0 and π = π . Give your answer to 4 decimal places.
By comparing β 1 + π₯ 2 to the average of π₯ and 1 + π₯ when π₯ > 0 , find better bounds for estimating πΏ . Give your answer to 4 decimal places.
Q3:
Let be the arc length of the polar curve over the interval . Express as a definite integral.
Using a calculator, or otherwise, find the value of giving your answer to 4 decimal places.
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