Arc length of cosh(x) on -1 to 1: finding the arc length of the hyperbolic cosine.
To find the arc length of the hyperbolic cosine function, we begin by recalling the arc length formula integral(sqrt(1+(f'(x))^2)dx on [a,b]. We take the derivative of cosh(x) to get sinh(x), so we're integrating sqrt(1+(sinh(x))^2) on [-1,1]. Fortunately, there's an identity relating sinh(x) and cosh(x): 1+sinh^2(x)=cosh^2(x), so the interior of the square root simplifies to cosh^2(x), and after taking the square root, we have the integral of cosh(x) on [-1,1].
This is a really interesting result! Apparently the arc length is equal to the area for the hyperbolic cosine function, and we're going to investigate this in more detail in a future video by systematically searching for all functions with this property. Is cosh(x) the only one?
Before we integrate, we use the fact that cosh(x) is an even function to cut the integration interval in half, producing a factor of 2 in front of the integral.
Now that we have the integral set up for the arc length of cosh(x) on -1 to 1, we finish finding the arc length by finding the antiderivative and evaluating across the limits of integration. The antiderivative of cosh(x) is sinh(x) and we evaluate from 0 to 1. We give a quick reminder of the definition of the hyperbolic sine function sinh(x)=1/2*(e^x-e^-x) and this means sinh(0)=0. Evaluating at the upper limit, we obtain an arc length of e-1/e, and we're done!
