扩展：“长度”是什么,是真的吗?

That was easy. We added up the lengths of each of the sides to get $1+1+1+1=4$.

Right! We know that the perimeter (or circumference for the case of a circle) was the radius times the diameter which is $42\pi$.

Well, those were easy because we knew the formulas for them. But what if we had the curve $y = x^2$ on the range $x \in [-1,1]$ and we wanted to find the "perimeter" of that curve? We don't have an immediate formula for it!

However, with the techniques you will learn in the upcoming section, you can not only find the perimeters of parabolas in given domains, but actually any curve! So read on.

Note. Be sure you are comfortable with integration before you read about lengths of curves.