Question

How do you find the arc length of the curve `y=lnx` from [1,5]?

Answer 1

The arc length is defined by the following formula: `L=int_1^5 sqrt{(x^2+1)/(x)}dx`

Explanation:

To find the arc length of a curve, we use the formula
`L=int_a^b sqrt{1+[f'(x)]^2}dx`

In our case, `f(x) = ln x`, `a=1`, and `b=5`.
The derivative of `f(x)=ln x` is `f'(x)=1/x`, so we get:

`L=int_1^5 sqrt{1+[1/x]^2}dx`
`L=int_1^5 sqrt{1+1/(x^2)}dx`
`L=int_1^5 sqrt{(x^2+1)/(x)}dx`