Suppose that we are given a function that is continuous on an interval and we want to calculate the length of the curve drawn out by the graph of from to . If the graph were a straight line this would be easy — the formula for the length of the line is given by Pythagoras' theorem. And if the graph were a piecewise linear function we can calculate the length by adding up the length of each piece.
The problem is that most graphs are not linear. Nevertheless we can estimate the length of the curve by approximating it with straight lines. Suppose the curve is given by the formula for . We divide the interval into subintervals with equal width and endpoints . Now let so is the point on the curve above . The length of the straight line between and is
So an estimate of the length of the curve is the sum
As we divide the interval into more pieces this gives a better estimate for the length of . In fact we make that a definition.
The Arclength Formula
Suppose that is continuous on . Then the length of the curve given by between and is given by
And in Leibniz notation
Proof: Consider . By the Mean Value Theorem there is a point in such that
Putting this into the definition of the length of gives
Now this is the definition of the integral of the function between and (notice that is continuous because we are assuming that is continuous). Hence
As a sanity check of our formula, let's calculate the length of the "curve" from to . First let's find the answer using the Pythagorean Theorem.
so the length of the curve, , is
Now let's use the formula
Arclength of a parametric curve
For a parametric curve, that is, a curve defined by and , the formula is slightly different:
Proof: The proof is analogous to the previous one: Consider and .
By the Mean Value Theorem there are points and in such that
Putting this into the definition of the length of the curve gives
This is equivalent to: