Question

How do you find the arc length of the curve `y=sqrt(x-3)` over the interval [3,10]?

Answer 1

Arc length will be approximately `7.716` units.

Explanation:

Recall that arc length of a curve is given by

`A = int_a^b sqrt(1 + (dy/dx)^2) dx`

The derivative of our curve is given by the chain rule as being `dy/dx= 1/(2sqrt(x -3))`.

`A = int_3^10 sqrt(1 + (1/(2sqrt(x- 3)))^2) dx`

`A = int_3^10 sqrt(1 + 1/(4(x - 3)))dx`

`A = int_3^10 sqrt((4x - 12 + 1)/(4(x -3)))dx`

`A = int_3^10 sqrt((4x - 11)/(4x - 12))dx`

This is a pretty complex integral, so I would solve using a graphing calculator.

Evaluating you should get `A = 7.716` units.

Hopefully this helps!