How do we find the length of the curve of $y=12/5 x+1$ from $x=0$ to $x=5$.

Question

How can I find the length of the curve: $$y=\frac{12}{5}x+1$$ from $x=0$ to $x=5$

I know the arc length formula is : $\ell = \int_a^b \sqrt{1 + (f'(x))^2} \, dx$

I'll need to set up the formula to look like: $\ell = \int_0^5 \sqrt{1+144/25} dx$

Is this correct?

Answer 1

While you can use the arc-length formula to compute the length of the curve (which in this case you can factor the $\sqrt{1 + 144/25} $ out of the integral since it's a constant), the easy non-calculus way to do this is just draw the curve (line) $y = \frac{12}{5}x + 1$ — put a horizontal line at the value $y(0) = 1$ and a vertical line at $x = 5$. Then computing the length of the curve is analogous to finding the length of the hypotenuse of the triangle with vertices $(0,1)$, $(5, 1)$, $(5, 13)$