9.9 Finding Arc Lengths of Curves Given by Polar Equations

• What does $r$ represent in polar coordinates?


  Distance from the origin
• What is the angle $\theta$ measured from in polar coordinates?


  Positive x-axis
• The relationship between Cartesian coordinates $(x, y)$ and polar coordinates $(r, \theta)$ is defined by the equations $x =$$r \cos \theta$ and y = r \sin \theta</latex>. The hidden word is coordinates
• The equation $x =$$r \sin \theta$ correctly describes the relationship between Cartesian and polar coordinates.


  False
• What are the Cartesian coordinates of the polar point (r, \theta) = (3, \frac{\pi}{4})</latex>?

  $\left(\frac{3\sqrt{2}}{2}, \frac{3\sqrt{2}}{2}\right)$
• The arc length $L$ of a polar curve defined by $r =$$f(\theta)$ from $\theta =$$a$ to $\theta =$$b$ is given by the formula $L =$ \int_{a}^{b} \sqrt{r^{2} + \left(\frac{dr}{d\theta}\right)^{2}} \, d\theta. The hidden word is arc
• What is the derivative of $r =$$\theta$ with respect to $\theta$?


  $\frac{dr}{d\theta} =$$1$
• The arc length of the polar curve $r =$$\theta$ from $\theta =$$0$ to $\theta =$$2\pi$ is approximately 21.256 units.
• Match the coordinate system with its arc length formula:

  Cartesian ↔️ $L =$$\int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^{2}} \, dx$

  Polar ↔️ $L =$ \int_{a}^{b} \sqrt{r^{2} + \left(\frac{dr}{d\theta}\right)^{2}} \, d\theta
• The arc length $L$ of a polar curve defined by $r =$$f(\theta)$ from $\theta =$$a$ to $\theta =$$b$ is given by the formula $L =$ \int_{a}^{b} \sqrt{r^{2} + \left(\frac{dr}{d\theta}\right)^{2}} \, d\theta. The hidden word is definite
• What is the approximate arc length of the polar curve $r =$$\theta$ from $\theta =$$0$ to $\theta =$$2\pi$?


  $21.256$
• What is the formula for the arc length of a polar curve defined by $r =$$f(\theta)$ from $\theta =$$a$ to $\theta =$$b$?


  $L =$ \int_{a}^{b} \sqrt{r^{2} + \left(\frac{dr}{d\theta}\right)^{2}} \, d\theta
• For $r =$$\theta$, the derivative $\frac{dr}{d\theta}$ is 1
• What is the approximate arc length of $r =$$\theta$ from $\theta =$$0$ to \theta = 2\pi</latex>?


  21.256
• The arc length formula for polar coordinates involves the derivative of $r$ with respect to $\theta$.
• Steps to find the arc length of a polar curve

  1️⃣ Express $r$ as a function of $\theta$

  2️⃣ Calculate $\frac{dr}{d\theta}$

  3️⃣ Substitute $r$ and $\frac{dr}{d\theta}$ into the arc length formula

  4️⃣ Integrate from $a$ to $b$
• What is the arc length of $r =$$2\cos(\theta)$ from $\theta =$$0$ to \theta = \frac{\pi}{2}</latex>?


  $\pi$
• For $r =$$2\cos(\theta)$, the derivative $\frac{dr}{d\theta}$ is -2\sin(\theta)
• The derivative of r = 2\cos(\theta)</latex> with respect to $\theta$ is $2\sin(\theta)$.


  False
• To find the arc length $L$ of a polar curve $r =$$f(\theta)$ from $\theta =$$a$ to $\theta =$$b$, use the formula
• What is the formula for the arc length of a polar curve $r =$$f(\theta)$ from $\theta =$$a$ to $\theta =$$b$?


  $L =$ \int_{a}^{b} \sqrt{r^{2} + \left(\frac{dr}{d\theta}\right)^{2}} \, d\theta
• To calculate $\frac{dr}{d\theta}$, differentiate $r$ with respect to \theta
• Steps to find the arc length of a polar curve

  1️⃣ Find $r$ as a function of $\theta$.

  2️⃣ Calculate $\frac{dr}{d\theta}$.

  3️⃣ Substitute $r$ and $\frac{dr}{d\theta}$ into the formula.

  4️⃣ Integrate from $a$ to $b$.
• Find the arc length of r = 2\cos(\theta)</latex> from $\theta =$$0$ to $\theta =$$\frac{\pi}{2}$.


  $\pi$
• The arc length of $r =$$2\cos(\theta)$ from $\theta =$$0$ to $\theta =$$\frac{\pi}{2}$ is $\pi$ units.
• Match the coordinate system with its arc length formula:

  Cartesian ↔️ $\int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^{2}} \, dx$

  Polar ↔️ \int_{a}^{b} \sqrt{r^{2} + \left(\frac{dr}{d\theta}\right)^{2}} \, d\theta
• What are the two components of polar coordinates?

  r</latex> and $\theta$
• In polar coordinates, $r$ represents the distance from the origin
• What is the formula to convert $x$ from polar to Cartesian coordinates?


  $x =$$r \cos \theta$
• Convert the polar coordinates $(4, \frac{\pi}{3})$ to Cartesian coordinates.


  $(2, 2\sqrt{3})$
• The arc length of $r =$$\theta$ from $\theta =$$0$ to $\theta =$$2\pi$ is approximately 21.256 units.
• Match the coordinate system with its arc length formula:

  Cartesian ↔️ L = \int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^{2}} \, dx</latex>

  Polar ↔️ $L =$ \int_{a}^{b} \sqrt{r^{2} + \left(\frac{dr}{d\theta}\right)^{2}} \, d\theta
• To derive the polar arc length formula, use the relationship between polar and Cartesian coordinates
• What is the simplified formula for the arc length of a polar curve after deriving it from Cartesian coordinates?

  $L =$ \int \sqrt{\left(\frac{dr}{d\theta}\right)^{2} + r^{2}} \, d\theta
• What is the simplified formula for the arc length of a polar curve?

  $L =$ \int \sqrt{\left(\frac{dr}{d\theta}\right)^{2} + r^{2}} \, d\theta
• The arc length of $r =$$\theta$ from $\theta =$$0$ to $\theta =$$2\pi$ is approximately 21.256
• The Cartesian arc length formula is L = \int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^{2}} \, dx</latex>.
• Match the coordinate system with its arc length formula:

  Cartesian ↔️ $L =$$\int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^{2}} \, dx$

  Polar ↔️ $L =$ \int_{a}^{b} \sqrt{r^{2} + \left(\frac{dr}{d\theta}\right)^{2}} \, d\theta
• What is the simplified formula for the arc length of a polar curve?

  $L =$ \int \sqrt{\left(\frac{dr}{d\theta}\right)^{2} + r^{2}} \, d\theta
• The arc length of $r =$$\theta$ from $\theta =$$0$ to $\theta =$$2\pi$ is approximately 21.256