Given the limaçon $r=1+2\cos \left(\theta \right)$, find the area that lies inside the larger loop and outside the smaller loop.

• A. $\frac{9\pi }{4}$
• B. $\frac{5\pi }{4}$
• C. $\frac{9\pi }{2}$
• D. $\frac{7\pi }{4}$

Answer: (A)

Verified step by step guidance

1. Identify the given limaçon equation: \(r = 1 + 2 \cos(\theta)\), which has both a larger loop and a smaller inner loop due to the coefficient of \(\cos(\theta)\) being greater than 1.
2. Find the values of \(\theta\) where the radius \(r\) equals zero to determine the points where the curve crosses the pole (origin). Solve \$1 + 2 \cos(\theta) = 0\( for \)\theta$.
3. Set up the integral expressions for the areas of the loops. The area enclosed by a polar curve from \(\theta = a\) to \(\theta = b\) is given by \(\frac{1}{2} \int_a^b r^2 \, d\theta\). Use this formula to write integrals for the larger loop and the smaller loop separately.
4. Determine the correct limits of integration for the larger loop and the smaller loop based on the \(\theta\) values found in step 2. The larger loop corresponds to the full range of \(\theta\) where \(r\) is positive and outside the smaller loop, while the smaller loop corresponds to the range where \(r\) is negative or inside the inner loop.
5. Calculate the area inside the larger loop and outside the smaller loop by subtracting the area of the smaller loop from the area of the larger loop using the integrals set up. This will give the desired area.