Arc Length

$Arc Length Example 1$

Arc length formula

The arc length of a function can be defined by the following formula:

$L=\int ds$ where

$ds=\int_a^b \sqrt{1+\left(\frac{dy}{dx}\right)^2}\ dx$ if $y=f(x)$

$ds=\int_a^b \sqrt{1+\left(\frac{dx}{dy}\right)^2}\ dy$ if $x=h(y)$

If you’re given a function in the form $y=f(x)$ , you’ll need to use the first formula for $ds$ and take the integral of that $ds$ in order to find the arc length of the function. Your limits of integration will need to be $x$ limits of integration. In the first $ds$ formula, $dy/dx$ will refer to the derivative of the function $y$ with respect to $x$ .

If you’re given a function in the form $x=h(y)$ , you’ll need to use the second formula for $ds$ and take the integral of that $ds$ in order to find arc length of the function. Your limits of integration will need to be $y$ limits of integration. In the second $ds$ formula, $dx/dy$ will refer to the derivative of the function $x$ with respect to $y$ .

Arc Length

Arc length formula

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