Find the arc length of the cardioid with polar equation .
Write the integral for the arc length of the spiral between and . Do not evaluate the integral.
The purpose of this question is to get improved estimates on the length of a spiral curve.
Use the fact that when to find lower and upper bounds for the length of the spiral between and . Give your answer to 4 decimal places.
By comparing to the average of and when , find better bounds for estimating . Give your answer to 4 decimal places.