What is the exact length of the spiraling polar curve r=5e^(2theta) from 0 to 2pi?

somi5fjt

somi5fjt

Answered question

2023-03-20

What is the exact length of the spiraling polar curve r = 5 e 2 θ from 0 to 2 π ?

Answer & Explanation

dalematealreypq3a

dalematealreypq3a

Beginner2023-03-21Added 10 answers

r ( θ ) = 5 e 2 θ   r ^
Arc length:
s = C s .   d t      = C v v   d t     
v ( θ ) = d d t ( 5 e 2 θ   r ^ )
Product rule:
= 10 e 2 θ θ .   r ^ + 5 e 2 θ d d t (   r ^ )     
d d t (   r ^ ) = d d t ( cos θ sin θ )
( - sin θ cos θ ) θ . = θ ^ θ .
Therefore, is:
v ( θ ) = 10 e 2 θ θ .   r ^ + 5 e 2 θ θ .   θ ^
And becomes:
= C ( 10 e 2 θ θ . ) 2 + ( 5 e 2 θ θ .   ) 2   d t
= C   e 2 θ 10 2 + 5 2        θ .   d t
= 5 5 0 2 π e 2 θ        d θ
= 5 5 2 ( e 4 π - 1 )

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?