To find the area of the inner loop of the limaçon $r = 1 + 2\cos\theta$, we’ll follow these steps:

1. Identify the range of $\theta$ that defines the inner loop.

   The inner loop appears when $r = 1 + 2\cos\theta$ is negative. To find when $r = 0$, set: $1 + 2\cos\theta = 0$ Solving for $\cos\theta$, we get: $\cos\theta = -\frac{1}{2}$ This happens at $\theta = \frac{2\pi}{3}$ and $\theta = \frac{4\pi}{3}$. Therefore, the inner loop is traced from $\theta = \frac{2\pi}{3}$ to $\theta = \frac{4\pi}{3}$.

2. Set up the area integral for the inner loop.

   The area $A$ inside a polar curve $r = f(\theta)$ from $\theta = a$ to $\theta = b$ is given by: $A = \frac{1}{2} \int_a^b r^2 \, d\theta$ For the inner loop, $a = \frac{2\pi}{3}$ and $b = \frac{4\pi}{3}$, so we have: $A = \frac{1}{2} \int_{\frac{2\pi}{3}}^{\frac{4\pi}{3}} (1 + 2\cos\theta)^2 \, d\theta$

3. Expand $(1 + 2\cos\theta)^2$ and simplify the integral.

   Expanding $(1 + 2\cos\theta)^2$ gives: $(1 + 2\cos\theta)^2 = 1 + 4\cos\theta + 4\cos^2\theta$ Using the identity $\cos^2\theta = \frac{1 + \cos(2\theta)}{2}$, we substitute: $1 + 4\cos\theta + 4\cos^2\theta = 1 + 4\cos\theta + 4\left(\frac{1 + \cos(2\theta)}{2}\right)$ Simplifying further: $= 1 + 4\cos\theta + 2 + 2\cos(2\theta) = 3 + 4\cos\theta + 2\cos(2\theta)$

4. Evaluate the integral.

   Substitute back into the integral: $A = \frac{1}{2} \int_{\frac{2\pi}{3}}^{\frac{4\pi}{3}} \left(3 + 4\cos\theta + 2\cos(2\theta)\right) d\theta$ We can now integrate each term separately over $\left[\frac{2\pi}{3}, \frac{4\pi}{3}\right]$:

   • For $\int_{\frac{2\pi}{3}}^{\frac{4\pi}{3}} 3 \, d\theta = 3\left(\frac{4\pi}{3} - \frac{2\pi}{3}\right) = 3 \cdot \frac{2\pi}{3} = 2\pi$.

   • The integrals of $\cos\theta$ and $\cos(2\theta)$ over this symmetric interval are zero, since they average out to zero over a complete cycle.

   Thus, we find: $A = \frac{1}{2} \cdot 2\pi = \pi$

Answer:

The area of the inner loop of the limaçon $r = 1 + 2\cos\theta$ is: $A = \pi$

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Relative Questions

1. How do you find the area of a polar curve in general?
2. Why does $r = 1 + 2\cos\theta$ produce an inner loop, and what does that mean geometrically?
3. What is the significance of symmetry in simplifying integrals over trigonometric functions?
4. Could the area of the outer loop for this limaçon be calculated in a similar way?
5. How would the solution change if the equation were $r = 1 + 2\sin\theta$?

Tip:

When integrating trigonometric functions over symmetric intervals, check if parts of the function (like $\cos \theta$ or $\cos(2\theta)$) average to zero—it can save time on calculations!