9.8 Finding Areas of Regions Bounded by Polar Curves

• What does $r$ represent in polar coordinates?


  Distance from the origin
• In polar coordinates, $\theta$ represents the angle from the positive x-axis to the line segment connecting the origin to the point
• A polar curve is defined by an equation in polar coordinates where $r$ is a function of $\theta$.
• What is the formula for finding the area of a region bounded by a polar curve?

  A = \frac{1}{2} \int_{\alpha}^{\beta} r^{2} \, d\theta</latex>
• In the area formula for polar curves, $\alpha$ and $\beta$ are the angles that define the boundaries of the region
• The area formula for polar curves is derived from the area of a circular sector.
• What type of polar curve is defined by the equation $r =$$2\cos(\theta)$?


  Circle
• The polar curve $r =$$2\cos(\theta)$ represents a circle with a diameter of 2 centered at (1, 0)
• Steps to find the area enclosed by the cardioid $r =$$1 +$$\cos(\theta)$


  1️⃣ Sketch the curve

  2️⃣ Determine limits of integration

  3️⃣ Apply the area formula

  4️⃣ Expand and integrate

  5️⃣ Evaluate the integral
• The cardioid $r =$$1 +$$\cos(\theta)$ is symmetric about the x-axis.
• The area enclosed by the cardioid $r =$$1 +$$\cos(\theta)$ is $\frac{3\pi}{2}$
• Match the polar curve with its area:

  Cardioid $r =$$1 +$$\cos(\theta)$ ↔️ $\frac{3\pi}{2}$

  Rose curve $r =$$3\cos(2\theta)$ ↔️ $\frac{9\pi}{8}$
• Steps to find the area inside one petal of the rose curve $r =$$3\cos(2\theta)$


  1️⃣ Sketch the curve

  2️⃣ Determine limits of integration

  3️⃣ Apply the area formula

  4️⃣ Expand and integrate

  5️⃣ Evaluate the integral
• The area inside one petal of the rose curve $r =$$3\cos(2\theta)$ is $\frac{9\pi}{8}$
• Over what range of $\theta$ is one petal of the rose curve r = 3\cos(2\theta)</latex> traced?


  $- \frac{\pi}{4} \to \frac{\pi}{4}$
• The general formula for finding the area of a region bounded by a single polar curve $r =$$f(\theta)$ is \frac{1}{2}
• What is the polar equation of the cardioid in Example 1?

  $r =$$1 +$$\cos(\theta)$
• The limits of integration for the entire cardioid $r =$$1 +$$\cos(\theta)$ are from \theta = 0</latex> to $\theta =$$2\pi$.
• Steps to find the area enclosed by the cardioid $r =$$1 +$$\cos(\theta)$.


  1️⃣ Apply the formula: $A =$$\frac{1}{2} \int_{0}^{2\pi} (1 + \cos(\theta))^{2} \, d\theta$

  2️⃣ Expand and integrate the integrand.

  3️⃣ Evaluate the definite integral.
• One petal of the rose curver = 3\cos(2\theta)</latex> is traced from $\theta =$$- \frac{\pi}{4}$ to $\theta =$$\frac{\pi}{4}$, which are the limits of integration.
• What is the area inside one petal of the rose curve $r =$$3\cos(2\theta)$?


  $\frac{9\pi}{8}$
• The formula for the area between two polar curves $r_{1} =$$f(\theta)$ and $r_{2} =$$g(\theta)$, where $f(\theta) \geq g(\theta)$, is $A =$$\frac{1}{2} \int_{\alpha}^{\beta} \left[f(\theta)^{2} - g(\theta)^{2}\right] \, d\theta$.
• To find the limits of integration for the area between two polar curves, one must solve for the angles where $f(\theta) =$$g(\theta)$, indicating the points of intersection.
• What are the limits of integration for finding the area between r_{1} = 2 + \cos(\theta)</latex> and $r_{2} =$$3\cos(\theta)$ from $\theta =$$0$ to $\theta =$$\frac{\pi}{2}$?


  $\theta =$$0$ to $\frac{\pi}{3}$
• The area between $r_{1} =$$2 +$$\cos(\theta)$ and $r_{2} =$$3\cos(\theta)$ from $\theta =$$0$ to $\frac{\pi}{3}$ is $\frac{\pi}{6} +$$\frac{\sqrt{3}}{4}$.
• What is the formula for finding the area between two polar curves r_{1} = f(\theta)</latex> and $r_{2} =$$g(\theta)$, where $f(\theta) \geq g(\theta)$?


  $A =$$\frac{1}{2} \int_{\alpha}^{\beta} \left[f(\theta)^{2} - g(\theta)^{2}\right] \, d\theta$
• The area formula in polar coordinates for a single curve $r =$$f(\theta)$ is derived from the area of a circular sector.
• In the area formula $A =$$\frac{1}{2} \int_{\alpha}^{\beta} r^{2} \, d\theta$, the term $\alpha$ represents the lower boundary of integration.
• What is the area enclosed by $r =$$2\cos(\theta)$ from $\theta =$$0$ to $\theta =$$\pi$?


  $\pi$
• Steps to find the area enclosed by the cardioid $r =$$1 +$$\cos(\theta)$ using polar coordinates.


  1️⃣ Determine the limits of integration: $\theta: 0 \to 2\pi$.

  2️⃣ Apply the area formula: $A =$$\frac{1}{2} \int_{0}^{2\pi} (1 + \cos(\theta))^{2} \, d\theta$.

  3️⃣ Expand and integrate the integrand.

  4️⃣ Evaluate the definite integral to find the area.
• Match the polar curve with its correct area:

  Cardioid $r =$$1 +$$\cos(\theta)$ ↔️ $\frac{3\pi}{2}$

  Rose Curve $r =$$3\cos(2\theta)$ ↔️ $\frac{9\pi}{8}$
• What are the limits of integration for the cardioid $r =$$1 +$$\cos(\theta)$?


  $0 \to 2\pi$
• What are the limits of integration for the rose curve $r =$$3\cos(2\theta)$?


  $- \frac{\pi}{4} \to \frac{\pi}{4}$
• The integral to find the area of the cardioid $r =$$1 +$$\cos(\theta)$ is A = \frac{1}{2} \int_{0}^{2\pi} (1 + \cos(\theta))^{2} \, d\theta</latex>
• The area of the cardioid $r =$$1 +$$\cos(\theta)$ is $\frac{3\pi}{2}$
• What formula is used to find the area between two polar curves $r_{1} =$$f(\theta)$ and $r_{2} =$$g(\theta)$?


  $A =$$\frac{1}{2} \int_{\alpha}^{\beta} \left[f(\theta)^{2} - g(\theta)^{2}\right] \, d\theta$
• What are the steps to find the area between two polar curves?

  1️⃣ Identify the curves and their equations

  2️⃣ Find the limits of integration

  3️⃣ Set up the integral

  4️⃣ Evaluate the integral
• The limits of integration are found by solving for the angles $\alpha$ and $\beta$ where the curves intersect
• When setting up the integral to find the area between two polar curves, the curve farther from the origin is squared first.
• What is the final step in finding the area between two polar curves?

  Evaluate the integral