In what sense is the length of a parameterized curve an area?

Question

A little confusion on my part. Study of multi variable calculus and we are using the formula for length of a parameterized curve. The equation makes intuitive sense and I can work it OK. But I also recall using the same integral with out the parameterizing to find the length of a curve where the first term of the square root in just one. The former formula is the general case.

Now for the question: I had just previously used the integral for completing the quadrature i.e. Find the area under a curve. Is the single integral used for finding both area and length ? I guess I am trying to unify the concepts in my mind to understand the context of how they are used and know the difference. Thank you.

Answer 1

You are given a function $\phi:\>[a,b]\to{\mathbb R}$ representing some "intensity" depending on $x$. The integral $$\int_a^b \phi(x)\>dx$$ then captures the "total impact" that this variable intensity has on the domain $[a,b]$.

If we are talking about the "area under a curve $y=f(x)\geq0$" then the intensity $\phi_{\rm area}(x)$ is just $f(x)$, because the area per "infinitesimal increment" $dx$ is $f(x)\,dx$. It follows that the total area under the curve between $x=a$ and $x=b$ is $$\int_a^b\phi_{\rm area}(x)\>dx=\int_a^b f(x)\>dx\ .$$ If we are talking about the "length of a curve $\gamma: \ y=f(x)$" (or some other geometrical or physical quantity associated to a curve $\gamma$) then the intensity $\phi_{\rm length}$ in question is $\sqrt{1+f'^2(x)}$ because a certain geometrical and limit argument has shown that the length increment of $\gamma$ per "infinitesimal increment" $dx$ is $\sqrt{1+f'^2(x)}\,dx$. It follows that the total length of the curve $\gamma$ is given by $$L(\gamma)=\int_a^b\phi_{\rm length}(x)\>dx=\int_a^b\sqrt{1+f'^2(x)}\>dx\ .$$

Answer 2

Yes. A single integral is the limit of a sequence of sums (having more and more terms) that approaches the quantity you want to find. You can find a curve’s length by using sums of the lengths of smaller and smaller straight-line segments between nearby points on the curve. You can find an area by using sums of the areas of narrower and narrower rectangles underneath the curve. Just what goes inside the integral depends on a little geometry, and there are assumptions that can be shown are sufficient to guarantee that the limit (the integral) does approach the thing you are calculating, but basically in both cases you are finding a quantity by using approximations that are sums and then taking a limit as the approximations get more and more accurate.