9.9 Finding Arc Lengths of Curves Given by Polar Equations

• What is the arc length formula in Cartesian coordinates?

  $\int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^{2}} dx$
• The arc length formula in polar coordinates is ∫
• The polar coordinate system uses radius and angle to define points.


  True
• In polar coordinates, the integral is evaluated with respect to θ
• What are the limits of integration for the polar equation $r =$$f(\theta)$ where $\alpha \leq \theta \leq \beta$?


  $\alpha$ and $\beta$
• What derivative is used in the arc length formula for polar coordinates?

  $\frac{dr}{d\theta}$
• What is the arc length formula in Cartesian coordinates?

  $\int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^{2}} dx$
• The arc length formula in polar coordinates uses `r` (radius) and θ
• The limits of integration in polar coordinates correspond to the curve's angle
• Steps to find the arc length in polar coordinates

  1️⃣ Identify the polar equation r = f(θ)

  2️⃣ Determine the angle range [α, β]

  3️⃣ Find the derivative dr/dθ

  4️⃣ Substitute into the arc length formula

  5️⃣ Evaluate the integral
• In polar coordinates, the limits of integration are angle values.
• Match the derivative with its coordinate system:

  Cartesian ↔️ dy/dx

  Polar ↔️ dr/dθ
• What is the arc length formula in polar coordinates?

  \int_{a}^{b} \sqrt{r^{2} + \left(\frac{dr}{d\theta}\right)^{2}} d\theta
• In polar coordinates, the limits of integration are angle values.

  True
• The arc length formula in polar coordinates is \int_{a}^{b} \sqrt{r^{2} + \left(\frac{dr}{d\theta}\right)^{2}} d\theta
• The Cartesian arc length formula uses x and y coordinates, while the polar arc length formula uses r and θ.


  True
• In Cartesian coordinates, the limits of integration for arc length are the x-values.
• If the polar equation is $r =$$f(\theta)$ where $\alpha \leq \theta \leq \beta$, the arc length is given by \int_{\alpha}^{\beta} \sqrt{r^{2} + \left(\frac{dr}{d\theta}\right)^{2}} d\theta
• If $r =$$2\theta +$$\cos(\theta)$, then $\frac{dr}{d\theta} =$$2 - \sin(\theta)$ because differentiation rules were applied.
• For $r =$$2\theta$ and $0 \leq \theta \leq \pi$, the arc length formula becomes \int_{0}^{\pi} \sqrt{(2\theta)^{2} + (2)^{2}} d\theta
• The limits of integration in the polar arc length formula depend on the angle values.


  True
• What is the simplified integrand for the arc length formula when $r =$$e^{\theta}$?


  $\sqrt{2}(e^{\theta})$
• Match the integration variable with its coordinate system:

  Cartesian ↔️ Integrates with respect to $x$

  Polar ↔️ Integrates with respect to $\theta$
• What is the arc length for r = e^{\theta}</latex> from $\theta =$$0$ to $2\pi$?


  $\sqrt{2}(e^{2\pi} - 1)$
• The arc length formula in Cartesian coordinates is ∫
• The arc length formula in polar coordinates uses the derivative $\frac{dr}{d\theta}$.


  True
• What is the arc length formula in polar coordinates?

  \int_{a}^{b} \sqrt{r^{2} + \left(\frac{dr}{d\theta}\right)^{2}} d\theta</latex>
• Match the coordinate system with its parameters:

  Cartesian ↔️ `x` and `y`

  Polar ↔️ `r` and `θ`
• In polar coordinates, the limits of integration correspond to angle values.


  True
• The key to determining the limits of integration in polar coordinates is to identify the appropriate angle range
• The arc length formula in polar coordinates requires the derivative of $r$ with respect to $\theta$.


  True
• In polar coordinates, the derivative used in the arc length formula is dy/dx.

  False
• Match the coordinate system with its derivative:

  Cartesian ↔️ dy/dx

  Polar ↔️ dr/dθ
• What is the arc length formula in polar coordinates when α ≤ θ ≤ β?

  \int_{\alpha}^{\beta} \sqrt{r^{2} + \left(\frac{dr}{d\theta}\right)^{2}} d\theta
• How are the limits of integration determined in Cartesian coordinates?

  x-values
• Why is finding dr/dθ necessary in polar coordinates for arc length calculations?

  It represents the rate of change of the radius with respect to the angle
• Determining dr/dθ is optional when calculating arc length in polar coordinates.

  False
• Polar coordinates use r (radius) and θ (angle).
• What is the arc length formula in Cartesian coordinates?

  $\int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^{2}} dx$
• Match the components of the arc length formulas with their descriptions:

  Cartesian coordinates ↔️ Uses x and y coordinates

  Polar coordinates ↔️ Uses r (radius) and θ (angle)

  Cartesian derivative ↔️ $\frac{dy}{dx}$

  Polar derivative ↔️ $\frac{dr}{d\theta}$