Sketch the region and use a double integral to find its area. The region bounded by the cardioid r= - 5(1 - cos 0)
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![## Calculating the Area Bounded by a Cardioid
**Objective:**
To sketch the region and use a double integral to find its area.
### Problem Statement:
Determine the area of the region bounded by the cardioid defined by the polar equation:
\[ r = -5(1 - \cos \theta) \]
### Steps to Solve:
1. **Sketch the Cardioid:**
- Begin by plotting the polar equation \( r = -5(1 - \cos \theta) \). Note that this cardioid is reflected due to the negative sign in front of the function.
2. **Set Up the Double Integral:**
- The area \( A \) of a region in polar coordinates can be determined using the double integral:
\[
A = \frac{1}{2} \int_{\theta_1}^{\theta_2} r^2 \, d\theta
\]
- For this cardioid, determine \( \theta_1 \) and \( \theta_2 \), the limits of integration that will fully trace the curve.
3. **Evaluate the Integral:**
- Substitute the given equation into the integral:
\[
A = \frac{1}{2} \int_{\theta_1}^{\theta_2} [-5(1 - \cos \theta)]^2 \, d\theta
\]
- Solve the integral to find the exact area enclosed by the cardioid.
4. **Interpret the Result:**
- The resulting area will represent the space enclosed by the cardioid in the polar coordinate system.
By following these steps, you can accurately determine the area enclosed within this particular cardioid using double integrals in polar coordinates.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8735dee1-ec31-4346-b70c-e4d10f64e04c%2F7f348b62-c483-4430-85e4-6a4f4976b730%2F5xekbqx.png&w=3840&q=75)
We have to find the area of the region bounded by the cardioid .
Using a graphing utility, sketch the polar curve and shade the region bounded by this cardioid as follows.
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