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Wize AP Calculus (BC) Textbook > Applications of Integration

Arc Length

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Arc Length

While the distance formula can tell us the length of a straight line, we can use integrals to find the length of a curve.


Arc Length of a Curve

Let f(x)f\left(x\right) be a function with continuous derivative function f′(x)f'\left(x\right) defined on the interval [a,b]\left[a,b\right]. The arc length of f(x)f\left(x\right) is given by
L=∫ab1+[f′(x)]2 dx\boxed{L=\int_a^b\sqrt{1+[f'\left(x\right)]^2} \ dx}

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Example: Arc Length

Find the length of the curve made by f(x)=−ln⁡(cos⁡x)f(x)=-\ln(\cos x) on [0,π4][0,\frac{\pi}{4}].

L=∫ab1+f′(x)2dx\displaystyle L=\int_a^b\sqrt{1+f'\left(x\right)^2}dx

f′(x)=−(−sin⁡xcos⁡x)=tan⁡x\displaystyle f'(x)=-\left(\frac{-\sin x}{\cos x}\right)= \tan x

=∫0π41+(tan⁡x)2dx=\displaystyle \int_0^\frac{\pi}{4}\sqrt{1+(\tan x)^2}dx

∫0π4sec⁡2xdx=∫0π4sec⁡xdx\displaystyle \int_0^\frac{\pi}{4} \sqrt{\sec^2 x}dx=\displaystyle \int_0^\frac{\pi}{4} \sec xdx

=ln⁡∣sec⁡x+tan⁡x∣∣0π4=\displaystyle \ln|\sec x + \tan x| \bigg|_0^\frac{\pi}{4}

=ln⁡∣sec⁡π4+tan⁡π4∣−ln⁡∣sec⁡0+tan⁡0∣∣0π4=\displaystyle \ln|\sec \frac{\pi}{4} + \tan \frac{\pi}{4}| -\displaystyle \ln|\sec 0 + \tan 0| \bigg|_0^\frac{\pi}{4}

=ln⁡∣2+1∣−ln⁡∣1+0∣=\displaystyle \ln|\sqrt2 + 1| -\displaystyle \ln|1 + 0|

=ln⁡∣2+1∣=\displaystyle \ln|\sqrt2 + 1|

Practice: Arc Length

Find the arc length of f(x)=12(ex+e−x)f(x)=\frac{1}{2}\left(e^x+e^{-x}\right) on [0,1][0,1].