Educator

Arclength

Main formula:

Hints and tips:

• To remember this formula, it helps to recall that it comes from the distance formula between two points, which in turn comes from the Pythagorean Theorem.

• Remember that you must integrate the square root formula above. A common mistake is to integrate the function itself, not the square root formula. Of course, this would give you the area under the curve and not the arclength.

• A similar mistake is to mix this up with formula for surface area of revolution, which looks similar. Be careful which one you are asked for.

• Don’t make the common algebraic mistake of thinking that reduces to a + b! This is extremely wrong, and your teacher will likely be merciless if you do it!

• Many problems in Calculus II classes are “rigged” so that when you expand 1 + f ′(x)² , it becomes a perfect square that cancels nicely with the square root.

• Often this perfect square is achieved by making the f ′(x)² be something of the form (a − b)² = a² − 2ab + b². Then the +1 changes it to a² + 2ab + b², which you can then factor as (a + b)².

• When it’s feasible, check that your answer makes sense. Unlike area integrals, which can be negative if a curve goes below the x-axis, arclength should always be positive! You might also be able to check that the curve looks about as long as your answer.