In this video we are going to talk about Arc Length. There is one main formula for arc length that one need to know and we are going to introduce it. We use formula for arc length when we are trying to find the length of a curve y=f(x) from point a to point b. Where this formula comes from is the Pythagorean Theorem. There is one common mistake that Calculus 2 students make. When they get an arc length problem they usually try to integrate the original function. Doing so, they get an area instead of the arc length. We will try out this formula on several examples.
You elegantly and simply explained this, thank you.
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.
Dont 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 its 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.