Find the length of the cardioid

r equals 2 plus 2 cosine thetar=2+2cosθ.

To find the length of a cardioid given in polar form as \( r = 2 + 2 \cos \theta \), we use the formula for the length \( L \) of a curve in polar coordinates:

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

For the cardioid \( r = 2 + 2 \cos \theta \), we first need to calculate \( \frac{dr}{d\theta} \).

### Step 1: Differentiate \( r \) with respect to \( \theta \)
\[
r = 2 + 2 \cos \theta
\]
\[
\frac{dr}{d\theta} = \frac{d}{d\theta} \left( 2 + 2 \cos \theta \right) = -2 \sin \theta
\]

### Step 2: Set up the integral for the curve length
Now we substitute \( r \) and \( \frac{dr}{d\theta} \) into the formula for \( L \):
\[
L = \int_0^{2\pi} \sqrt{(2 + 2 \cos \theta)^2 + (-2 \sin \theta)^2} \, d\theta
\]

Simplify the expression inside the square root:
\[
L = \int_0^{2\pi} \sqrt{(2 + 2 \cos \theta)^2 + 4 \sin^2 \theta} \, d\theta
\]
\[
= \int_0^{2\pi} \sqrt{4 + 8 \cos \theta + 4 \cos^2 \theta + 4 \sin^2 \theta} \, d\theta
\]
\[
= \int_0^{2\pi} \sqrt{4 + 8 \cos \theta + 4 (\cos^2 \theta + \sin^2 \theta)} \, d\theta
\]
Since \( \cos^2 \theta + \sin^2 \theta = 1 \), the expression simplifies to:
\[
L = \int_0^{2\pi} \sqrt{8 + 8 \cos \theta} \, d\theta
\]
Factor out the common term:
\[
L = \int_0^{2\pi} \sqrt{8(1 + \cos \theta)} \, d\theta = \int_0^{2\pi} 2\sqrt{2} \sqrt{1 + \cos \theta} \, d\theta
\]

### Step 3: Use the identity for \( 1 + \cos \theta \)
We can simplify \( \sqrt{1 + \cos \theta} \) using the trigonometric identity:
\[
1 + \cos \theta = 2 \cos^2 \left( \frac{\theta}{2} \right)
\]
Thus, the integral becomes:
\[
L = \int_0^{2\pi} 2\sqrt{2} \cdot \sqrt{2} \cos \left( \frac{\theta}{2} \right) \, d\theta
\]
\[
L = 4 \int_0^{2\pi} \cos \left( \frac{\theta}{2} \right) \, d\theta
\]

### Step 4: Evaluate the integral
The integral of \( \cos \left( \frac{\theta}{2} \right) \) over \( [0, 2\pi] \) gives:
\[
L = 4 \cdot \left[ 2 \right] = 8
\]

Thus, the length of the cardioid is \( \boxed{8} \).

Would you like more details or have any questions?

### Related Questions:
1. How is the length formula for a polar curve derived?
2. What are the differences between a cardioid and other polar curves?
3. How would the length change if the cardioid equation were modified?
4. Can you find the area enclosed by the cardioid \( r = 2 + 2 \cos \theta \)?
5. How would the length be computed for more complex polar curves?

### Tip:
Always simplify the integrand as much as possible before evaluating the integral to avoid complications.

Find the length of the cardioid r = 2 + 2cosθ.

Learn how to calculate the length of the cardioid r = 2 + 2cosθ using polar coordinates and integral calculus. This detailed step-by-step solution covers key trigonometric identities and curve length formulas. Suitable for advanced high school or undergraduate students.

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