Chapter 6: Q. 84 (page 541)

Prove the surprising fact that the arc length of the catenary curve traced out by the hyperbolic function f(x) = cosh x on any interval [a,b] is equal to the area under the same graph on [a,b].

Short Answer

Expert verified

Hence, proved that the arc length of the catenary curve traced out by the hyperbolic function f(x) = cosh x on any interval [a, b] is equal to the area under the same graph on [a, b].

Step by step solution

Step 1. To prove

The surprising fact that the arc length of the catenary curve traced out by the hyperbolic function f(x) = cosh x on any interval [a,b] is equal to the area under the same graph on [a,b].

Step 2. Differentiate the function with respect to x

Recall that the arc length of a differentiable function f(x)with continuous derivative from x = a

to x=b is given by the definite integral

$L = \int_{b}^{a} \sqrt{1 + \sin {(h)}^{2} d x}$

Note that the hyperbolic function f(x) = cosh x is differentiable and has continuous derivative on any interval [a,b]. So, differentiate the function with respect to x

f(x) = cosh x

f(x)=sinh x

Substitute the above derivative in the integral given in equation (1) and evaluate the integral

$L = \int_{b}^{a} \sqrt{1 + \sin {(h)}^{2} d x} = \int_{a}^{b} \cos h x d x = {[\sin h x]}^{b}_{a} = \sin h b - \sin h a$

So, the arc length of the catenary on the interval is sinh b-sinh a.

Next, find the area of catenary under the same graph on [a, b] . Remember that the area of a curve bound between the graph of the function and x-axis on [a, b] is given by the definite integral

$A = \int_{a}^{b} f (x) d x Thus the area under the graph of the catenary f (x) = \cosh x between x = a and x = b is given A = \int_{a}^{b} \cosh xdx = {[sinhx]}_{a}^{b} = sinhb - \sin ha$