# Length of Arc in XY-Plane | Applications of Integration

Apply Pythagorean theorem to the triangular strip shown in the figure:

$(ds)^2 = (dx)^2 + (dy)^2$ ← Equation (1)

Divide both sides of Equation (1) by (dx)^{2}

$\dfrac{(ds)^2}{(dx)^2} = \dfrac{(dx)^2}{(dx)^2} + \dfrac{(dy)^2}{(dx)^2}$

$\left( \dfrac{ds}{dx} \right)^2 = 1 + \left( \dfrac{dy}{dx} \right)^2$

$\dfrac{ds}{dx} = \sqrt{1 + \left( \dfrac{dy}{dx} \right)^2}$

$ds = \sqrt{1 + \left( \dfrac{dy}{dx} \right)^2} \, dx$

Integrate both sides

Divide both sides of Equation (1) by (dy)^{2}

$\dfrac{(ds)^2}{(dy)^2} = \dfrac{(dx)^2}{(dy)^2} + \dfrac{(dy)^2}{(dy)^2}$

$\left( \dfrac{ds}{dy} \right)^2 = \left( \dfrac{dx}{dy} \right)^2 + 1$

$\dfrac{ds}{dy} = \sqrt{\left( \dfrac{dx}{dy} \right)^2 + 1}$

$ds = \sqrt{\left( \dfrac{dx}{dy} \right)^2 + 1} ~ dy$

Integrate both sides