How do you find the length of the polar curve #r=cos^3(theta/3)#?

Answer 1

Use the chain rule.

By Chain Rule,

#{dr}/{d theta}=3cos^2(theta/3)cdot[-sin(theta/3)]cdot1/3#

by cleaning up a bit,

#=-cos^2(theta/3)sin(theta/3)#

Let us first look at the curve #r=cos^3(theta/3)#, which looks like this:

Note that #theta# goes from #0# to #3pi# to complete the loop once.

Let us now find the length #L# of the curve.

#L=int_0^{3pi}sqrt{r^2+({dr}/{d theta})^2} d theta#

#=int_0^{3pi}sqrt{cos^6(theta/3)+cos^4(theta/3)sin^2(theta/3)}d theta#

by pulling #cos^2(theta/3)# out of the square-root,

#=int_0^{3pi}cos^2(theta/3)sqrt{cos^2(theta/3)+sin^2(theta/3)}d theta#

by #cos^2theta=1/2(1+cos2theta)# and #cos^2theta+sin^2theta=1#,

#=1/2int_0^{3pi}[1+cos({2theta}/3)]d theta#

#=1/2[theta+3/2sin({2theta}/3)]_0^{3pi}#

#=1/2[3pi+0-(0+0)]={3pi}/2#

I hope that this was helpful.

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 length of the polar curve ( r = \cos^3\left(\frac{\theta}{3}\right) ), you can use the arc length formula for polar curves:

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

where ( r ) is the function that defines the curve, and ( \alpha ) and ( \beta ) are the initial and final angles, respectively.

In this case, ( r = \cos^3\left(\frac{\theta}{3}\right) ), so we need to find ( \frac{{dr}}{{d\theta}} ) and then integrate from ( \alpha ) to ( \beta ), the range of ( \theta ).

[ \frac{{dr}}{{d\theta}} = \frac{{d}}{{d\theta}}\left(\cos^3\left(\frac{\theta}{3}\right)\right) ]

Apply the chain rule to differentiate ( \cos^3\left(\frac{\theta}{3}\right) ).

[ \frac{{dr}}{{d\theta}} = -\frac{1}{3}\sin\left(\frac{\theta}{3}\right)\cos^2\left(\frac{\theta}{3}\right) ]

Now, plug ( r ) and ( \frac{{dr}}{{d\theta}} ) into the arc length formula and integrate from the initial angle ( \alpha ) to the final angle ( \beta ).

[ L = \int_{\alpha}^{\beta} \sqrt{\cos^6\left(\frac{\theta}{3}\right) + \frac{1}{9}\sin^2\left(\frac{\theta}{3}\right)\cos^4\left(\frac{\theta}{3}\right)} , d\theta ]

After integration, you will get the length of the polar curve within the given range of angles.

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