Calculus of a Single Variable, Chapter 7, 7.4, Section 7.4, Problem 13

The arc length of the curve y=f(x) between x=a and x=b, (aL=int_a^b sqrt(1+y'^2)dx
Before we start using the above formula let us notice that y=1/2(e^x+e^-x)=cosh x. This should simplify our calculations.
L=int_0^2sqrt(1+(cosh x)'^2)dx=
int_0^2sqrt(1+sinh^2x )dx=
Now we use the formula cosh^2 x=1+sinh^2x.
int_0^2cosh x dx=
sinh x|_0^2=
sinh 2-sinh 0=
sinh 2 approx3.62686
The arc length of the graph of the given function over interval [0,2] is sinh 2 or approximately 3.62686.
The graph of the function can be seen in the image below (part of the graph for which we calculated the length is blue).

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