Parametrization technique in the line integral

Question

Suppose I have

$$\int_C \,(3x+y)\,\mathbb dx+(2y-x)\,\mathbb dy$$

and I want to find the line integral from the straight line $(0,1)\mapsto (2,5)$.

Most of the things I see on YouTube or some sites, most of them using parametrization $t$ with fixed interval $0<t<1$.

E.g. for my problem:

$$x : 0\mapsto 2 \Rightarrow x=2t+0\quad 0\leq t \leq 1$$

$$y : 1\mapsto 5 \Rightarrow y=4t+1\quad 0\leq t \leq 1.$$

What is the secret behind $0\leq t \leq 1$?

Then, substituting in the integral we have the following :

$$\int_0^1 \,(3(2t)+(4t+1))\,2\mathbb dt+(2(4t+1)-(2t))\,4\mathbb dt$$

Could you explain to me? Why we always have interval $0\leq t \leq 1$?

Or, it's just a coincidence?

Thanks.

Answer 1

It might be useful to rewrite the parametrization a bit: $$ (x,y) = (0,1) + t(2,4). $$ The interpretation goes as follows: at $t=0$, the starting point is $(0,1)$. After $t$ seconds, one has moved $t$ steps in the direction $(2,4)$. So after one second $t=1$, we arrive at the end point $$ (0,1) + (2,4) = (2,5). $$ Therefore, $t$ lies between $0$ and $1$.

Answer 2

It's not "just a coincidence" --- it's a trick folks have learned. You can certainly do the parameterization in many other ways as well. Here's the big idea:

If you write $$ y = (1-t)A + tB $$ then when $t = 0$, you'll have $y = A$, and when $t = 1$, you'll have $y = B$. THat's because the functions $(1-t)$ and $t$ take on particularly nice easy values (zeros and ones!) at $t = 0$ or $t = 1$, so the formula above is really easy to remember.

The cool thing is that you don't have to use individual numbers for $A$ and $B$. You could even write points, and say that $$ t \mapsto (1-t)(0, 1) + t(2, 5) $$ which we interpret as \begin{align} t &\mapsto (1-t)(0, 1) + t(2, 5)\\ & = (0(1-t), 1(1-t)) + (2t, 5t)\\ & = (0, 1-t) + (2t, 5t)\\ & = (2t, 1-t + 5t)\\ & = (2t, 1+4t) \end{align} gives a path that runs from $(0, 1)$ to $(2, 5)$ as $t$ ranges from $0$ to $1$.

And eventually, you'll learn that $A$ and $B$ can even be things like functions, and folks write things like $$ H(x, t) = (1-t) \sin(x) + t \cos(x) $$ as a way to build a function which, as $t$ varies between $0$ and $1$, varies between $\sin x$ and $\cos x$. Fortunately, that's still a little ways away from where you are now.