9.3 Finding Arc Lengths of Curves Given by Parametric Equations

• Parametric equations use a parameter called t
• The parametric arc length formula is derived by substituting `dx = f'(t) dt` and `dy = g'(t) dt` into the infinitesimal arc length formula `ds = \sqrt{(dx)^2 + (dy)^2}`.True
• What is an example of parametric equations that cannot be easily expressed using a Cartesian equation?

  $x =$$t^{2}, y =$$t^{3}$
• The arc length of a curve defined by parametric equations is calculated using the formula \int_{a}^{b} \sqrt{(f'(t))^{2} + (g'(t))^{2}} dt
• The parametric equations x = t² and y = t³ define a curve that cannot be easily expressed using a Cartesian
• What are the derivatives of the parametric equations x = 2t and y = 3t²?

  f'(t) = 2, g'(t) = 6t
• Steps to find the arc length of a curve defined by parametric equations

  1️⃣ Find the derivatives f'(t) and g'(t).

  2️⃣ Substitute the derivatives into the formula.

  3️⃣ Evaluate the integral over the given range of t.
• The parametric arc length formula involves integrating the square root of the sum of squared derivatives.


  True
• Plugging the derivatives into the parametric arc length formula, we get $\int_{0}^{1} \sqrt{4 + 36t^{2}} dt$
• Match the feature with the type of equation:

  Single equation `y = f(x)` ↔️ Cartesian equation

  Two or more equations `x = f(t)`, `y = g(t)` ↔️ Parametric equation
• What is an example of a curve that cannot be easily expressed using a Cartesian equation but can be defined parametrically?

  x = t^2, y = t^3
• The first step in finding the arc length is to calculate the derivatives
• The flexibility of parametric equations allows for representing curves that cannot be easily expressed using a single Cartesian equation.
• Parametric equations use a parameter, typically denoted as t
• Parametric equations can represent curves that are difficult to describe using Cartesian equations.
• What is the formula to find the arc length of a curve defined by parametric equations?

  \int_{a}^{b} \sqrt{(f'(t))^{2} + (g'(t))^{2}} dt
• The parametric arc length formula allows for more flexibility in representing curves compared to the arc length formula for Cartesian equations.


  True
• The Cartesian arc length formula is applicable for curves that can be expressed as $y =$$f(x)$.


  True
• Evaluating definite integrals is a key step in finding the arc length of a curve defined by parametric equations.

  True
• The first step in finding the arc length is to calculate the derivatives
• The arc length formula for Cartesian equations uses the variable $x$.


  True
• The arc length of a curve defined by trigonometric parametric equations $x =$$a \cos t$ and $y =$$a \sin t$ from $t =$$0$ to $t =$$2\pi$ is $2\pi a$
• What do mixed parametric equations combine to define a curve?

  Parametric and Cartesian equations
• What is the purpose of simplifying the integral in the arc length formula?

  Reduce computational complexity
• What is the formula for finding the arc length of a curve defined by parametric equations?

  \int_{a}^{b} \sqrt{(f'(t))^{2} + (g'(t))^{2}} dt
• Parametric equations are less flexible than Cartesian equations for expressing curves.

  False
• Why is the parametric arc length formula more flexible than the Cartesian arc length formula?

  No single function required
• To find the arc length, the derivatives must be substituted into the parametric arc length formula and the integral evaluated.


  True
• Parametric equations allow for more flexibility in representing curves compared to Cartesian equations.


  True
• The key advantage of parametric equations is their flexibility in representing curves that cannot be easily expressed using a single Cartesian
• What are the derivatives of the parametric equations x = 2t and y = 3t^2?

  f'(t) = 2, g'(t) = 6t
• What is the parameter typically used in parametric equations?

  t
• The parametric arc length formula allows for curves that cannot be expressed as a single Cartesian function.


  True
• What formula is used to find the arc length of a curve defined by parametric equations?

  \int_{a}^{b} \sqrt{(f'(t))^{2} + (g'(t))^{2}} dt
• What is the difference between parametric and Cartesian equations?

  Parametric uses parameters
• Steps to find the arc length using the parametric arc length formula

  1️⃣ Find the derivatives f'(t) and g'(t)

  2️⃣ Substitute the derivatives into the formula

  3️⃣ Evaluate the definite integral
• Match the formula type with its variables:

  Parametric arc length ↔️ $t$

  Cartesian arc length ↔️ $x$
• Match the equation type with its definition:

  Cartesian equation ↔️ Single equation $y =$$f(x)$

  Parametric equation ↔️ Two or more equations $x =$$f(t)$, $y =$$g(t)$
• Steps to find the arc length of a curve defined by parametric equations $x =$$f(t)$ and $y =$$g(t)$


  1️⃣ Find the derivatives: Calculate $f'(t)$ and $g'(t)$.

  2️⃣ Substitute the derivatives into the formula: \int_{a}^{b} \sqrt{(f'(t))^{2} + (g'(t))^{2}} dt.

  3️⃣ Evaluate the definite integral over the given range of the parameter $t$.
• What variable is used in the parametric arc length formula?

  $t$