# What is the general equation for the arclength of a line?

##### 2 Answers

If we wish to find the arc length of

#### Explanation:

The general equation of a line is

Recall the formula for arc length is

The derivative of the linear function is

#A = int_a^b sqrt(1 + m^2)dx#

#A = [sqrt(1+ m^2)x]_a^b#

#A = bsqrt(1 + m^2) - asqrt(1 + m^2)#

#A = (b - a)sqrt(1 + m^2)#

Now let's verify to see if our formula is correct. Let

#A = (6 - 2)sqrt(1 + 2^2) = 4sqrt(5)#

If we were to use pythagoras, by connecting a horizontal line to a vertical line, we would get the following"

#y(2) = 5#

#y(6) = 13#

#Delta y = 13 - 5 = 8#

#Delta x = 4#

Thus

#A = sqrt(80) = sqrt(16 * 5) = 4sqrt(5)#

As obtained using our formula.

Hopefully this helps!

#### Explanation:

For the arc length of a linear function given its slope

Let

This may look scary because of all of the variables, but

The antiderivative is