# Thread: Limacon, Inner Loop Area

1. ## Limacon, Inner Loop Area

Find the area inside the loop of the following limacon: r = 7 − 14 sin (θ).

2. Originally Posted by Del
Find the area inside the loop of the following limacon: r = 7 − 14 sin (θ).

to find the limits of integration we set r =0

$0=7-14\sin(\theta) \iff \sin(\theta)=\frac{1}{2}$

so we get $\frac{\pi}{6} \mbox{ and } \frac{5 \pi}{6}$

$\frac{1}{2}\int_{\pi/6}^{5\pi/6}(7-14\sin(\theta))^2d\theta$

After integrating I get $49\pi-\frac{147\sqrt{3}}{2} \approx 26.63$

3. Originally Posted by Del
Find the area inside the loop of the following limacon: r = 7 − 14 sin (θ).

did you draw the graph?

now that we have drawn it, we see that the inner loop happens when the graph goes to the origin, that is, when $r = 0$. so let's solve for that:

$\Rightarrow 0 = 7 - 14 \sin \theta$

$\Rightarrow \sin \theta = \frac 12$

$\Rightarrow \theta = \frac {\pi}6,~\frac {5 \pi}6$ for $0 \le \theta \le 2 \pi$

thus, the area is given by: $A = \frac 12 \int_{\pi /6}^{5 \pi /6}r^2~d \theta = \frac 12 \int_{\pi / 6}^{5 \pi / 6}(7 - 14 \sin \theta)^2 ~d \theta$

EDIT: Geez, too late. Thanks a lot EmptySet!

4. We can factor out the 7 and write it as $7(1-2sin{\theta})$

$1-2sin{\theta}=0, \;\ {\theta}=\frac{\pi}{6}, \;\ {\theta}=\frac{5\pi}{6}$

Then, we get $\frac{49}{2}\int_{\frac{\pi}{6}}^{\frac{5\pi}{6}}( 1-2sin{\theta})^{2}d{\theta}$

$\frac{49}{2}\int_{\frac{\pi}{6}}^{\frac{5\pi}{6}}\ left[4sin^{2}{\theta}-4sin{\theta}+1\right]d{\theta}$

Oops, beat to the punch. At least we agree.

5. Originally Posted by galactus
EDIT ... Well, we agree. That's good.
Yes, the one benefit to posting after others have posted.

6. I didn't just copy what you all did. What would be the point in that.

I was in the middle of it when you fellas posted.

7. Originally Posted by galactus
I didn't just copy what you all did. What would be the point in that.

I was in the middle of it when you fellas posted.
of course not. no one suggested that.

but it's good that we all agree. our answers confirm one another. i mean, what are the chances we all got it wrong ...

8. I'm sorry, I misunderstood what you meant.

9. Originally Posted by galactus
I'm sorry, I misunderstood what you meant.
that's alright. no hard feelings whatsoever

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# looped limacons geometry

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