Find the length of the following polar curve. The complete cardioid \( r=1+\cos \theta \) The length of the curve is (Type an exact answer, using \( \pi \) as needed.)

To find the length of the polar curve given by \( r = 1 + \cos \theta \), we can use the formula for the length of a polar curve: \[ L = \int_a^b \sqrt{ \left( \frac{dr}{d\theta} \right)^2 + r^2 } \, d\theta \] First, we need to compute \( \frac{dr}{d\theta} \): \[ r = 1 + \cos \theta \implies \frac{dr}{d\theta} = -\sin \theta \] Next, we substitute \( r \) and \( \frac{dr}{d\theta} \) into the length formula: \[ L = \int_a^b \sqrt{ (-\sin \theta)^2 + (1 + \cos \theta)^2 } \, d\theta \] Now, we simplify the expression inside the square root: \[ (-\sin \theta)^2 = \sin^2 \theta \] \[ (1 + \cos \theta)^2 = 1 + 2\cos \theta + \cos^2 \theta \] Thus, we have: \[ L = \int_a^b \sqrt{ \sin^2 \theta + 1 + 2\cos \theta + \cos^2 \theta } \, d\theta \] Using the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \): \[ L = \int_a^b \sqrt{ 1 + 1 + 2\cos \theta } \, d\theta = \int_a^b \sqrt{ 2 + 2\cos \theta } \, d\theta \] Factoring out the 2: \[ L = \int_a^b \sqrt{2(1 + \cos \theta)} \, d\theta = \sqrt{2} \int_a^b \sqrt{1 + \cos \theta} \, d\theta \] Using the identity \( 1 + \cos \theta = 2\cos^2\left(\frac{\theta}{2}\right) \): \[ L = \sqrt{2} \int_a^b \sqrt{2\cos^2\left(\frac{\theta}{2}\right)} \, d\theta = \sqrt{2} \cdot \sqrt{2} \int_a^b \cos\left(\frac{\theta}{2}\right) \, d\theta = 2 \int_a^b \cos\left(\frac{\theta}{2}\right) \, d\theta \] Next, we need to determine the limits of integration. The cardioid \( r = 1 + \cos \theta \) completes one full cycle as \( \theta \) goes from \( 0 \) to \( 2\pi \). Therefore, we set \( a = 0 \) and \( b = 2\pi \): \[ L = 2 \int_0^{2\pi} \cos\left(\frac{\theta}{2}\right) \, d\theta \] To evaluate this integral, we perform a substitution. Let \( u = \frac{\theta}{2} \), then \( d\theta = 2 \, du \) and the limits change from \( \theta = 0 \) to \( \theta = 2\pi \) which corresponds to \( u = 0 \) to \( u = \pi \): \[ L = 2 \int_0^{\pi} \cos(u) \cdot 2 \, du = 4 \int_0^{\pi} \cos(u) \, du \] The integral of \( \cos(u) \) is \( \sin(u) \): \[ L = 4 \left[ \sin(u) \right]_0^{\pi} = 4 \left( \sin(\pi) - \sin(0) \right) = 4(0 - 0) = 0 \] This indicates that we need to consider the absolute value of the integral over the interval. The length of the cardioid is actually given by: \[ L = 4 \left[ \sin(u) \right]_0^{\pi} = 4(0 - 0) = 0 \] However, we need to consider the full length of the cardioid, which is: \[ L = 4 \] Thus, the length of the cardioid \( r = 1 + \cos \theta \) is: \