9.3 Finding Arc Lengths of Curves Given by Parametric Equations

• What is the arc length formula for $y =$$x^{2}$ from $x =$$0$ to $x =$$1$?


  $L =$$\int_{0}^{1} \sqrt{1 + 4x^{2}} \, dx$
• The arc length of $y =$$x^{2}$ from $x =$$0$ to $x =$$1$ is approximately $1.1478$
• The arc length of a curve defined by parametric equations is calculated using the formula L = \int_{a}^{b} \sqrt{\left(\frac{dx}{dt}\right)^{2} + \left(\frac{dy}{dt}\right)^{2}} \, dt</latex>, where $[a, b]$ is the interval of the parameter t
• What is the derivative of $x =$$3t$ with respect to $t$?


  $3$
• Trigonometric substitution is used to solve the integral $\int_{0}^{2} \sqrt{9 + 16t^{2}} \, dt$
• The arc length formula for parametric equations sums the infinitesimal arc lengths as the parameter $t$ varies from $a$ to b
• What are parametric equations used to define?

  A curve
• The parametric equations $x =$$2t$ and $y =$$t^{2}$ trace a parabola as $t$ changes.
• What does the parameter $t$ in parametric equations describe?


  How coordinates change
• The formula for arc length in Cartesian coordinates involves the derivative $\frac{dy}{dx}$.
• The arc length formula in Cartesian coordinates is L = \int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^{2}} \, dx</latex>, where $a$ and $b$ are the limits of integration
• What is the derivative $\frac{dy}{dx}$ for the function $y =$$x^{2}$?


  $2x$
• The arc length formula in parametric form uses the derivatives \frac{dx}{dt}</latex> and $\frac{dy}{dt}$.
• The arc length formula in parametric form is $L =$ \int_{a}^{b} \sqrt{\left(\frac{dx}{dt}\right)^{2} + \left(\frac{dy}{dt}\right)^{2}} \, dt, where $a$ and $b$ are the limits of the parameter
• What is the derivative $\frac{dx}{dt}$ for the parametric equationx = t</latex>?


  $1$
• Match the parametric equations with their derivatives:

  $x =$$3t$ ↔️ $\frac{dx}{dt} =$$3$

  $y =$$2t^{2}$ ↔️ $\frac{dy}{dt} =$$4t$
• The derivative of $y =$$x^{2}$ is 2x
• What is the first step in calculating the arc length of a curve defined by parametric equations?

  Find the derivatives
• The arc length of the curve $x =$$3t$ and $y =$$2t^{2}$ for 0 \le t \le 2</latex> is approximately $10.803$
• What are the parametric equations describing the position of a satellite orbiting Earth?

  $x(t) =$$5\cos(t)$, $y(t) =$$4\sin(t)$
• To find the total distance a satellite covers in one orbit, you calculate the arc length.
• The arc length of the satellite's orbit is approximately 28.64</latex> units.
• Match the parametric equations with their derivatives:

  $x =$$2t$ ↔️ $\frac{dx}{dt} =$$2$

  $y =$$3t^{2}$ ↔️ $\frac{dy}{dt} =$$6t$

  $x =$$t^{3}$ ↔️ $\frac{dx}{dt} =$$3t^{2}$
• What is the purpose of parametric equations in defining a curve?

  Define coordinates using a parameter
• The arc length formula in Cartesian coordinates is $L =$$\int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^{2}} \, dx$
• For $y =$$x^{2}$, the derivative $\frac{dy}{dx}$ is 2x
• What is the formula for arc length using parametric equations?

  L = \int_{a}^{b} \sqrt{\left(\frac{dx}{dt}\right)^{2} + \left(\frac{dy}{dt}\right)^{2}} \, dt</latex>
• Steps to calculate arc length using parametric equations

  1️⃣ Find the derivatives $\frac{dx}{dt}$ and $\frac{dy}{dt}$

  2️⃣ Substitute into the arc length formula

  3️⃣ Evaluate the integral
• The arc length formula for parametric equations $x =$$f(t)$ and $y =$$g(t)$ over an interval $[a, b]$ is given by: L
• The arc length formula sums infinitesimal arc lengths obtained from the derivatives of $x$ and $y$ with respect to $t$.
• What are the parametric equations used in the first example to calculate arc length?

  x = t</latex> and $y =$$t^{2}$
• The exact arc length of the curve $x =$$t$ and $y =$$t^{2}$ from $t =$$0$ to $t =$$1$ is approximately 1.1478.
• What are the parametric equations used in the second example to calculate arc length?

  $x =$$\cos(t)$ and $y =$$\sin(t)$
• The arc length of the curve $x =$$\cos(t)$ and $y =$$\sin(t)$ from $t =$$0$ to $t =$$\pi$ is equal to $\pi$.
• The simplified integrand for the AUV problem is: $\sqrt{25 + 25t^{2}}$
• What is the approximate arc length traveled by the AUV along its spiral path?

  $46.47$ units
• The arc length of the curve $x(t) =$$t^{2}$ and $y(t) =$$t^{3}$ from $t =$$0$ to $t =$$1$ is approximately 1.44.
• The arc length of the circle defined by $x(t) =$$4\cos(t)$ and y(t) = 4\sin(t)</latex> for $0 \le t \le 2\pi$ is approximately $8\pi$