Question

How do you find the arc length of the curve `y = 2x - 3`, `-2 ≤ x ≤ 1`?

Answer 1

The arc length is `3sqrt{5}\approx 6.7082`

Explanation:

Since the graph of `y=f(x)=2x-3` is a straight line, there's actually no need to use calculus. Instead, just find the straight-line distance between the points `(-2,f(-2))=(-2,-7)` and `(1,f(1))=(1,-1)`. The answer, by the distance formula (Pythagorean theorem), is

`sqrt{(-2-1)^2+(-7-(-1))^2}=sqrt{9+36}=sqrt{45}`

`=sqrt{9}sqrt{5}=3sqrt{5}\approx 6.7082`

If you want to confirm this with calculus, evaluate the integral `int_{-2}^{1}sqrt{1+(f'(x))^2}\ dx=int_{-2}^{1}sqrt{1+4}\ dx`

`=sqrt{5}x|_{-2}^{1}=sqrt{5}(1-(-2))=3sqrt{5}`