Lesson: Length of a Polar Curve

In this lesson, we will learn how to use the polar arc length formula for a parametric curve to find its length.

Worksheet: Length of a Polar Curve • 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 = 3 𝜃 over the interval 0 𝜃 𝜋 2 . Express 𝑠 as a definite integral.

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

Q4:

Find the total arc length of 𝑟 = 3 𝜃 s i n .

Q5:

Find the arc length of the polar curve , where lies in the interval .

Q6:

Consider the polar curve , where lies in the interval . Find a definite integral that represents the arc length of this curve.

Q7:

Consider the polar curve , where lies in the interval . Find a definite integral that represents the arc length of this curve.

Q8:

Find the arc length of the polar curve 𝑟 = 5 𝜃 , where 𝜃 lies in the interval [ 0 , 2 𝜋 ] .

Q9:

Find a definite integral that represents the arc length of 𝑟 = 1 + 𝜃 s i n on the interval 0 𝜃 2 𝜋 .

Q10:

Find the arc length of the polar curve given by 𝑟 = 𝑒 3 𝜃 on the interval 0 𝜃 2 .

Q11:

Find the arc length of the polar curve 𝑟 = 1 𝜃 s i n over the interval 0 𝜃 2 𝜋 .

Q12:

Find the arc length of the polar curve , where lies in the interval .

Q13:

Consider the polar curve , where lies in the interval . Find a definite integral that represents the arc length of this curve.

Q14:

Find the arc length of the polar curve , where lies in the interval .

Q15:

Find the arc length of the cardioid 𝑟 = 2 + 2 𝜃 c o s .

Q16:

Consider the polar curve , where lies in the interval . Find a definite integral that represents the arc length of this curve.

Q17:

Find the arc length of the polar curve 𝑟 = 8 + 8 𝜃 c o s over the interval 0 𝜃 𝜋 .

Q18:

Let 𝑠 be the arc length of the polar curve 𝑟 = 2 𝜃 2 over the interval 0 𝜃 𝜋 . Express 𝑠 as a definite integral.

Using a calculator, or otherwise, find the value of 𝑠 giving your answer to 3 decimal places.

Q19:

Let 𝑠 be the arc length of the polar curve 𝑟 = 𝜃 2 s i n 2 over the interval 0 𝜃 𝜋 . Express 𝑠 as a definite integral.

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

Q20:

Let 𝑠 be the arc length of the polar curve 𝑟 = 2 𝜃 over the interval 𝜋 𝜃 2 𝜋 . Express 𝑠 as a definite integral.

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

Q21:

Find a definite integral that represents the arc length of 𝑟 = 4 𝜃 c o s in the interval 0 𝜃 𝜋 2 .

Q22:

Find the arc length of the polar curve , where lies in the interval .

Q23:

Find the arc length of the polar curve , where lies in the interval .

Q24:

Find a definite integral that represents the arc length of 𝑟 = 𝑒 𝜃 on the interval 0 𝜃 1 .

Q25:

Consider the polar curve , where lies in the interval . Find a definite integral that represents the arc length of this curve.

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