8.15 Arc Length of a Curve

• What does the term "arc length" refer to?

  Length of a curve
• For a curve in non-parametric form, we use the derivative dy/dx
• For parametric curves, we use the derivatives dx/dt and dy/dt to calculate arc length.


  True
• What theorem is used in deriving the arc length formula?

  Pythagorean theorem
• The arc length formula is derived using the fundamental theorem of calculus.


  True
• What is the formula for ds^2 using the Pythagorean theorem?

  $ds^{2} =$$dx^{2} +$$dy^{2}$
• Steps to apply the arc length formula for y = f(x):

  1️⃣ Compute dy/dx

  2️⃣ Square the derivative

  3️⃣ Add 1 to the square of the derivative

  4️⃣ Take the square root

  5️⃣ Integrate from a to b
• The parametric form arc length formula uses the derivatives $\frac{dx}{dt}$ and $\frac{dy}{dt}$


  True
• Match the curve form with its formula and derivatives:

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

  Parametric ↔️ \int_{a}^{b} \sqrt{\left(\frac{dx}{dt}\right)^{2} + \left(\frac{dy}{dt}\right)^{2}} \, dt ||| $\frac{dx}{dt}$, $\frac{dy}{dt}$
• What is the first step in applying the arc length formula for a non-parametric curve `y = f(x)`?

  Compute the derivative
• For parametric curves, we use the derivatives `dx/dt` and `dy/dt` instead of `dy/dx` in the arc length formula.


  True
• Match the form of the curve with its arc length formula:

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

  Parametric ↔️ \int_{a}^{b} \sqrt{\left(\frac{dx}{dt}\right)^{2} + \left(\frac{dy}{dt}\right)^{2}} \, dt
• What are the two forms of curves for which arc length can be calculated using definite integrals?

  Non-parametric and parametric
• For a non-parametric curve, the arc length formula uses the derivative $\frac{dy}{dx}$.


  True
• If $x =$$t^{2}$ and $y =$$t^{3}$, then \frac{dx}{dt}</latex> is 2t
• The parametric arc length formula involves integrating the square root of the sum of the squares of $\frac{dx}{dt}$ and $\frac{dy}{dt}$ with respect to t
• The arc length of $x =$$t^{2}$, $y =$$t^{3}$ from $t =$$0$ to $t =$$1$ is approximately 1.4397


  True
• Steps for using basic substitution to calculate arc length

  1️⃣ Identify a composite function in the integrand

  2️⃣ Substitute a part of the integrand with a new variable

  3️⃣ Find the derivative of the new variable

  4️⃣ Rewrite the integral in terms of the new variable

  5️⃣ Evaluate the simplified integral
• The formula for integration by parts is $\int u \, dv =$$uv - \int v \, du$, where $v$ is the integral of dv
• When using basic substitution, the differential dx must be rewritten in terms of du.


  True
• Match the expression with the correct trigonometric substitution:

  $\sqrt{a^{2} - x^{2}}$ ↔️ $x =$$a\sin\theta$

  \sqrt{a^{2} + x^{2}} ↔️ $x =$$a\tan\theta$

  $\sqrt{x^{2} - a^{2}}$ ↔️ $x =$$a\sec\theta$
• The derivative of $u =$$9x^{4}$ with respect to $x$ is $\frac{du}{dx} =$$36x^{3}$.


  True
• The arc length of $y =$$x^{3}$ from $x =$$0$ to $x =$$2$ can be calculated using the substitution $u =$$9x^{4}$.


  True
• What is the length of the parabolic cable described by $y =$$0.01x^{2}$ from $x =$$- 50$ to $x =$$50$?


  $\approx 100.333$ meters
• Arc length is calculated using definite integrals.

  True
• Match the formula with the type of curve:

  Arc Length = $\int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^{2}} \, dx$ ↔️ Non-parametric form

  Arc Length = \int_{a}^{b} \sqrt{\left(\frac{dx}{dt}\right)^{2} + \left(\frac{dy}{dt}\right)^{2}} \, dt ↔️ Parametric form
• To derive the arc length formula, we use concepts from differential geometry
• Steps in deriving the arc length formula for a non-parametric curve:

  1️⃣ Consider an infinitesimal arc length ds

  2️⃣ Use the Pythagorean theorem

  3️⃣ Substitute dy = f'(x) dx

  4️⃣ Simplify the expression

  5️⃣ Take the square root and integrate
• For parametric curves, we use the derivatives dx/dt and dy/dt
• The expression ds^2 = dx^2 + (f'(x) dx)^2 simplifies to ds^2 = (1 + (f'(x))^2) dx^2.


  True
• Steps to apply the arc length formula for a curve $y =$$f(x)$ between points $a$ and $b$


  1️⃣ Compute the derivative $\frac{dy}{dx}$

  2️⃣ Square the derivative $\left(\frac{dy}{dx}\right)^{2}$

  3️⃣ Add 1 to the square of the derivative $1 +$$\left(\frac{dy}{dx}\right)^{2}$

  4️⃣ Take the square root $\sqrt{1 + \left(\frac{dy}{dx}\right)^{2}}$

  5️⃣ Integrate with respect to $x$ from $a$ to $b$
• For the non-parametric example $y =$$x^{2}$, the derivative $\frac{dy}{dx}$ is $2x$


  True
• The arc length formula for a non-parametric curve is derived using the Pythagorean theorem and infinitesimal arc length concepts.


  True
• The arc length formula for a non-parametric curve is \int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^{2}} \, dx
• Steps to apply the arc length formula for a non-parametric curve `y = f(x)`:

  1️⃣ Compute the derivative $\frac{dy}{dx}$

  2️⃣ Square the derivative $\left(\frac{dy}{dx}\right)^{2}$

  3️⃣ Add 1 to the square of the derivative: $1 +$$\left(\frac{dy}{dx}\right)^{2}$

  4️⃣ Take the square root of the sum: $\sqrt{1 + \left(\frac{dy}{dx}\right)^{2}}$

  5️⃣ Integrate the result with respect to `x` from `a` to `b`:
• What are the derivatives used in the arc length formula for parametric curves?

  $\frac{dx}{dt}$ and $\frac{dy}{dt}$
• The arc length formula for a parametric curve involves integrating with respect to the variable `t`.


  True
• The arc length formula for a parametric curve is \int_{a}^{b} \sqrt{\left(\frac{dx}{dt}\right)^{2} + \left(\frac{dy}{dt}\right)^{2}} \, dt
• What is the derivative of $y =$$x^{2}$?


  $2x$
• What is the value of $\frac{dy}{dt}$ if $y =$$t^{3}$?


  $3t^{2}$