What is the arclength of the polar curve #f(theta) = 2thetasin(5theta)-thetacot2theta # over #theta in [pi/12,pi/2] #?

Answer 1

#infty#

For this solution, I am assuming that your #f# is actually referring to the radial component.
If we travel an infinitely small angle, the distance will be f(#theta#) d#theta#. Therefore, we can integrate that function across that whole range to get the value, i.e. #int_(pi/12)^(pi/2) f(theta)d theta #
However, since cotangent goes to infinity very quickly at #pi/2#, this integral does not converge to a finite value, hence the length is infinite.
Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 2

To find the arc length of the polar curve ( f(\theta) = 2\theta \sin(5\theta) - \theta \cot(2\theta) ) over ( \theta ) in ( \left[ \frac{\pi}{12}, \frac{\pi}{2} \right] ), use the formula for arc length in polar coordinates:

[ L = \int_{\theta_1}^{\theta_2} \sqrt{r^2 + \left(\frac{dr}{d\theta}\right)^2} , d\theta ]

where ( r = f(\theta) ) and ( \frac{dr}{d\theta} ) is the derivative of ( f(\theta) ) with respect to ( \theta ).

Compute ( \frac{dr}{d\theta} ) by differentiating ( f(\theta) ) with respect to ( \theta ), then plug it into the formula along with ( r = f(\theta) ).

Evaluate the integral over the given interval ( \left[ \frac{\pi}{12}, \frac{\pi}{2} \right] ).

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7