8.15 Arc Length of a Curve

• Arc length is the distance along a curve between two points
• Arc length allows for the accurate measurement of complex curves using integration.
• The length of a curve between $x =$$a$ and $x =$$b$ can be calculated using the arc length formula, which involves integrating the square root of $1 +$$\left(\frac{dy}{dx}\right)^{2}$ with respect to $x$ from $a$ to $b$.a
• What is the expression for the differential arc length $ds$ in terms of $dx$ and $\frac{dy}{dx}$?


  $ds =$$\sqrt{1 + \left(\frac{dy}{dx}\right)^{2}} dx$
• The arc length formula includes the derivative of the function with respect to $x$.
• Arc length is the distance along a curve between two points
• The arc length of the curve $y =$$x^{2}$ from $x =$$0$ to x = 2</latex> can be found using integration.
• What is the formula for calculating arc length in Cartesian coordinates?

  $L =$$\int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^{2}} dx$
• Steps to calculate the arc length of $y =$$x^{2}$ from $x =$$0$ to x =2</latex>


  1️⃣ Find $\frac{dy}{dx}$: $\frac{dy}{dx} =$$2x$

  2️⃣ Substitute into the arc length formula: $L =$$\int_{0}^{2} \sqrt{1 + (2x)^{2}} dx$

  3️⃣ Evaluate the integral: $L \approx 4.6468$
• To apply the arc length formula, we first need to find $\frac{dy}{dx}$ for the given function
• What does $\frac{dy}{dx}$ represent?


  The derivative of $y$
• The arc length formula is L = \int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^{2}} dx</latex>, where $\frac{dy}{dx}$ is the derivative of the function.
• To apply the arc length formula, we first need to find $\frac{dy}{dx}$ for the given function.
• What is the first step in applying the arc length formula?

  Find $\frac{dy}{dx}$
• After finding $\frac{dy}{dx}$, we substitute it into the arc length formula.
• The arc length formula requires the derivative $\frac{dy}{dx}$ to be squared.
• What is the derivative of $y =$$x^{3}$ with respect to $x$?


  3x^{2}</latex>
• For $y =$$x^{3}$, the arc length formula becomes $L =$$\int_{0}^{1} \sqrt{1 + (3x^{2})^{2}} dx$, where $(3x^{2})^{2}$ is the square of the derivative.
• What is the derivative of $y =$$x^{3}$ with respect to $x$?


  $3x^{2}$
• Steps involved in applying the arc length formula

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

  2️⃣ Substitute into the arc length formula

  3️⃣ Evaluate the resulting integral
• The arc length formula can be applied to various functions by following the same steps.
• What are two common integration techniques used in arc length problems?

  U-substitution and trigonometric substitution
• U-substitution simplifies the integral when the integrand contains a function and its derivative.
• To find the arc length of $y =$$x^{2}$ from $x =$$0$ to $x =$$2$, we can use U-substitution.
• What substitution should be used for \int \sqrt{1 + (2x)^{2}} dx</latex>?

  $u =$$2x$
• To solve arc length problems involving integration, we typically use U-substitution or trigonometric substitution to evaluate the integrals
• When the integrand contains a function and its derivative (or a constant multiple of the derivative), U-substitution simplifies the integral
• Steps to find the arc length of $y =$$x^{2}$ from $x =$$0$ to $x =$$2$ using U-substitution


  1️⃣ Find $\frac{dy}{dx}$: $2x$

  2️⃣ Substitute into the arc length formula: $L =$$\int_{0}^{2} \sqrt{1 + (2x)^{2}} dx$

  3️⃣ Let $u =$$2x$, then $du =$$2 dx$

  4️⃣ Rewrite the integral: $L =$$\frac{1}{2} \int \sqrt{1 + u^{2}} du$

  5️⃣ Use a standard integral formula

  6️⃣ Substitute back $u =$$2x$ and evaluate from $x =$$0$ to $x =$$2$
• U-substitution is effective when the integrand contains a function and its derivative.
• When the integrand contains square roots of quadratic expressions, trigonometric substitution is an effective method
• Trigonometric substitution is used when the integrand contains expressions like $\sqrt{a^{2} - x^{2}}$, \sqrt{a^{2} + x^{2}}, or $\sqrt{x^{2} - a^{2}}$.
• Arc length is significant in calculus because it allows us to accurately measure the length of complex curves
• What is the formula for arc length in Cartesian coordinates?

  L = \int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^{2}} dx</latex>
• Steps to find the arc length of $y =$$x^{2}$ from $x =$$0$ to $x =$$2$ using the arc length formula


  1️⃣ Find $\frac{dy}{dx}$: $2x$

  2️⃣ Substitute into the arc length formula: $L =$$\int_{0}^{2} \sqrt{1 + (2x)^{2}} dx$

  3️⃣ Evaluate the integral
• The arc length formula is L = \int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^{2}} dx</latex>, where $L$ is the arc length, $a$ and $b$ are the limits of integration
• What does $\frac{dy}{dx}$ represent in the arc length formula?


  Derivative of y with respect to x
• To apply the arc length formula, we first need to find $\frac{dy}{dx}$ for the given function
• Steps to find the arc length of $y =$$x^{3}$ from $x =$$0$ to $x =$$1$


  1️⃣ Find $\frac{dy}{dx}$: $3x^{2}$

  2️⃣ Substitute into the arc length formula: $L =$$\int_{0}^{1} \sqrt{1 + (3x^{2})^{2}} dx$

  3️⃣ Evaluate the integral
• U-substitution and trigonometric substitution are commonly used to evaluate arc length integrals.
• What is the first step in finding the arc length of $y =$$\sin x$ from $x =$$0$ to $x =$$\frac{\pi}{2}$?


  $\frac{dy}{dx} =$$\cos x$