Find the exact length of the arc of this curve

Question

$y = 2e^x + (1/8)e^{-x}$ ... on the interval $[0, \ln(2)]$

I know am supposed to user the Arc Length formula, but I'm not sure if I have the derivative of this function correct.

I came up with:

$$f'^2 (x)= 4e^{2x} - \frac{1}{2} + \frac{1}{64} e^{-2x}$$

I'm really rusty on this stuff though and am probably wrong.

And even if this is right, I'm not sure what to do next with all that + 1 under the square root.

Answer 1

hint:$$1+(f'(x))^2 = \left(2e^x + \frac{1}{8}e^{-x}\right)^2$$

Answer 2

You can use the formulae:

$$\int_a^b \sqrt{y'(x)^2+1} \, dx$$

This is arclength for cartesian coordinates. Evaluate derivative $y'$, square it and add one, should be:

$1+\left(2 e^x-\frac{e^{-x}}{8}\right)^2=\left(2 e^x+\frac{e^{-x}}{8}\right)^2$

Now use formulae from above.