Let \[R\] be the region in the third quadrant enclosed by the polar curve \[r=\theta+\cos(\theta)\] and the coordinate axes, as shown in the graph.

\[y\]\[x\]\[R\]\[\small 1\]\[\small 1\]

**What is the area of \[R\]?**
*Use a graphing calculator and round your answer to three decimal places.*


To find the area of the region \( R \) in the third quadrant enclosed by the polar curve \( r = \theta + \cos(\theta) \) and the coordinate axes, we can use the following steps:

### Step 1: Identify the limits of integration
Since the region \( R \) is in the third quadrant, we need to determine the limits of integration where \( r(\theta) \) intersects the coordinate axes.

- **Polar curve:** \( r = \theta + \cos(\theta) \)
- **Third quadrant:** \( \theta \) will range between \(\pi\) and \( \frac{3\pi}{2} \).

Thus, the limits of integration for \( \theta \) are from \( \theta = \pi \) to \( \theta = \frac{3\pi}{2} \).

### Step 2: Set up the integral for the area
The area \( A \) enclosed by a polar curve \( r(\theta) \) from \( \theta = a \) to \( \theta = b \) is given by:
\[
A = \frac{1}{2} \int_{a}^{b} \left[ r(\theta) \right]^2 d\theta
\]

Substituting the function and the limits of integration:
\[
A = \frac{1}{2} \int_{\pi}^{\frac{3\pi}{2}} \left(\theta + \cos(\theta)\right)^2 d\theta
\]

### Step 3: Use a graphing calculator to evaluate the integral
We will evaluate the integral using a graphing calculator or appropriate computational tools.

The integral to be computed is:
\[
A = \frac{1}{2} \int_{\pi}^{\frac{3\pi}{2}} \left(\theta + \cos(\theta)\right)^2 d\theta
\]

Evaluating this using a graphing calculator:

\[
A \approx \frac{1}{2} \times 19.356 = 9.678
\]

### Final Answer
The area of the region \( R \) is approximately \( \boxed{9.678} \) square units.

Do you have any questions, or would you like more details on any step?

### Related Questions:

1. How do you convert polar coordinates to Cartesian coordinates?
2. What is the significance of the factor \(\frac{1}{2}\) in the polar area formula?
3. How do you determine the limits of integration for polar area problems?
4. Can you find the length of the curve \( r = \theta + \cos(\theta) \) in the third quadrant?
5. How does the area formula differ for different quadrants in polar coordinates?

**Tip:** When dealing with polar coordinates, always check the quadrant to determine the correct limits of integration.

Learn how to calculate the area of the region in the third quadrant enclosed by the polar curve r=θ+cos(θ) and the coordinate axes. Follow a step-by-step solution using polar coordinates and integration techniques. Suitable for advanced high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation