9.8 Finding Areas of Regions Bounded by Polar Curves

• Polar coordinates use horizontal and vertical distances, similar to Cartesian coordinates.

  False
• The radial distance \( r \) in polar coordinates is calculated using the formula \( r = \sqrt{x^2 + y^2} \), which relates it to Cartesian coordinates.
• In polar coordinates, a point is represented using an angle and a distance from the origin.
• Parametric equations in polar coordinates use two functions: $r(t)$ for distance and $\theta(t)$ for the angle.
• Parametric equations in polar coordinates are useful for representing non-single-valued functions.
• Cartesian coordinates are best suited for representing linear paths.
• Parametric equations in polar coordinates use two functions: $r(t)$ for distance and $\theta(t)$ for the angle.
• Polar coordinates are particularly useful for expressing curves like the cardioid and rose, which are difficult to represent in Cartesian form.


  True
• What parameter is used in parametric equations in polar coordinates?

  $t$
• Match the form of polar equations with their variables:

  Standard Polar ↔️ $r, \theta$

  Parametric Polar ↔️ $r, \theta, t$
• The curve defined by $r(t) =$$t$ and $\theta(t) =$$2t$ for $0 \leq t \leq 2\pi$ is a spiral-like curve
• The area formula in polar coordinates is equivalent to the area formula in Cartesian coordinates.

  False
• Polar coordinates define a point in a plane using the distance from the origin and the angle
• Match the conversion formula with its purpose:

  x = r cos(θ) ↔️ Convert polar to Cartesian x

  y = r sin(θ) ↔️ Convert polar to Cartesian y

  r = √(x^2 + y^2) ↔️ Convert Cartesian to polar r

  θ = arctan(y/x) ↔️ Convert Cartesian to polar θ
• Polar coordinates are best suited for representing circular and spiral shapes.


  True
• Polar coordinates can represent curves that are difficult to express in Cartesian form.


  True
• A curve defined by $r(t) =$$t$ and $\theta(t) =$$2t$ forms a spiral-like shape.


  True
• Polar coordinates use an angle and a distance from the origin to define a point.


  True
• Polar coordinates are useful for representing curves like the cardioid, rose, and lemniscate.


  True
• What variables are used in polar coordinates?

  $r, \theta$
• Parametric equations in polar coordinates use two functions, $r(t)$ and $\theta(t)$, to define the distance and the angle
• Parametric equations can represent non-single-valued functions, which standard polar equations cannot.


  True
• What type of curves are best represented using parametric polar equations?

  Complex curves
• What is the formula for calculating the area of a region bounded by a polar curve?

  $A =$$\frac{1}{2} \int_{\alpha}^{\beta} [f(\theta)]^{2} \, d\theta$
• What is the area inside the cardioid $r =$$1 +$$\cos(\theta)$ for 0 \leq \theta \leq 2\pi</latex>?


  $\frac{3\pi}{2}$
• What are the limits of integration for finding the area of a region bounded by a polar curve?

  $\alpha, \beta$
• The polar area formula is particularly useful for calculating the area of regions bounded by circular and curvilinear shapes.
• Polar coordinates use angle and distance from the origin, while Cartesian coordinates use horizontal and vertical distances.


  True
• Polar coordinates represent a point using an angle and a distance from the origin.
• The formula to convert y from polar to Cartesian coordinates is $y =$$r\sin\theta$.
• Cartesian coordinates are ideal for representing linear paths.

  True
• Match the polar form with its variables, representation, and use cases:

  Standard Polar ↔️ $r, \theta$ || $r =$$f(\theta)$ || Simple curves, single-valued functions

  Parametric Polar ↔️ $r, \theta, t$ || $r =$$r(t), \theta =$$\theta(t)$ || Complex curves, motion analysis
• In the polar area formula, $\alpha$ and $\beta$ represent the starting and ending angles of the region.
• What is the area formula for polar coordinates?

  $\frac{1}{2} \int_{\alpha}^{\beta} [f(\theta)]^{2} \, d\theta$
• What do $\alpha$ and $\beta$ represent in the polar area formula?


  Starting and ending angles
• What is a key advantage of using the polar area formula?

  Curvilinear shapes are easier
• The Fundamental Theorem of Calculus states that if $F(x)$ is an antiderivative of $f(x)$, then \int_{a}^{b} f(x) \, dx = F(b) - F(a)
• What is the antiderivative of $x^{2}$?


  $F(x) =$$\frac{1}{3}x^{3}$
• What is the formula to find the area between two polar curves $r_{1}$ and $r_{2}$?


  $A =$$\frac{1}{2} \int_{\alpha}^{\beta} [f_{1}(\theta)^{2} - f_{2}(\theta)^{2}] \, d\theta$
• Steps to find the area between two polar curves

  1️⃣ Identify the intersection points

  2️⃣ Determine the outer and inner curves

  3️⃣ Set up the integral

  4️⃣ Evaluate the integral