Question

What is the arclength of `f(x)=x-sqrt(e^x-2lnx)` on `x in [1,2]`?

Answer 1

`L=int_1^2sqrt(1+(1-(xe^x-2)/(2xsqrt(e^x-2lnx)))^2)dxapprox1.0630`

Explanation:

The arc length of the curve of `f` on `x in [a,b]` is given by

`L=int_a^bsqrt(1+(f'(x))^2)dx`

Here, `f(x)=x-(e^x-2lnx)^(1/2)` so

`f'(x)=1-1/2(e^x-2lnx)^(-1/2)(e^x-2/x)`

`color(white)(f'(x))=1-(xe^x-2)/(2xsqrt(e^x-2lnx))`

Then the arc length is given by

`L=int_1^2sqrt(1+(1-(xe^x-2)/(2xsqrt(e^x-2lnx)))^2)dxapprox1.0630`