# What is the meaning of the area bound by two curves?

As the title states, I don't fully understand what the meaning of the area between two curves is, in an application sense. For example, if I was provided with the equations of two velocity curves, what does the area between some region from a to b mean? Is it the distance between the two objects in motion?

Edit: Thanks to everyone that responded to the post, it was all very useful :D

• Welcome to stackexchange. When the curves represent velocities then that area is indeed the distance between the particles at time $b$ (assuming they started at the same place at time $a$). So you do in fact understand the meaning better than you think you do (and better than many beginning calculus students). – Ethan Bolker Nov 30 '17 at 16:50

I would say the meaning of "the area between two curves" in a context such as the one in the question is literally that you have two functions, each of which has been plotted as a curve in a Cartesian plane with the horizontal axis appropriately labeled so that it is the input variable of both functions; if that variable is $t,$ then the area between the two curves from $a$ to $b$ means you draw vertical lines $t=a$ and $t=b,$ and now the two curves and the two lines enclose a region in the plane, and we measure the area of this region.

But people also use the phrase "area between the curves" as a kind of verbal metaphor and/or graphical shorthand for a definite integral of the difference of two functions. The basis of this metaphor is that in many cases we can compute the area bounded by the graphs of two functions and by two vertical lines by taking a definite integral of the difference of those two functions. The problem with the metaphor is that if we integrate $f(t) - g(t)$ over an interval on which $f(t)$ is sometimes less than $g(t),$ we end up with regions of the plane whose area has to be subtracted from the total "area" in order to have the "area" come out equal to the integral.

And finally there is the application of "the area between two curves," which often comes down to knowing that there is one process of some sort that is increasing a quantity at a certain rate and another process that is decreasing the same quantity at a certain rate, and the "area" (actually the integral of the difference of those two rates integrated over a certain period of time) is the accumulated effect of those two processes.

So if you have two objects moving along the same straight line, starting from the same point at time $a,$ and the velocity of each object is is how fast it is moving to the right, then if the velocity of object number $1$ is given by $v_1(t)$ and the velocity of object number $2$ is given by $v_2(t),$ the integral $$\int_a^b (v_1(t) - v_2(t))\, dt$$ tells me how much farther to the right object number $1$ will be compared to object $2$ at time $b.$ This also happens to be numerically equal to the area between the graphs of the two velocity functions from $a$ to $b,$ with the caveats that this works only if we chose all our units appropriately and that if $v_1$ ever dips below $v_2$ on that graph then we need to treat parts of the area between the two curves as "negative area."

In an application like that, my viewpoint is that what people really want to do is to integrate the difference of two functions (which of course is itself a function), and the phrase "area between the curves" is just their way of saying that they would like to visualize this integral by plotting the graphs of those two functions on a Cartesian plane.

If you have a graph of velocity vs. time for a particle in motion, that is, the y-axis denotes the velocity and the x-axis denotes time, then the area under the graph tells you the displacement of the particle. Then, the area between two graphs will tell you the difference in displacement between the initial positions and final positions of the particles.

Actually it represents the following quantities

1. Relative displacement if you are considering signed area, that is $$r_{1|2}=\int(v_1(t) - v_2(t)) dt$$

2. (Edit) The following integral (the area between the two plots) does not represent any formal quantity:

$$\int|v_{1} (t) - v_2 (t)| dt$$

Note that the distance between the particles is represented as:

$$d = \left|\int(v_1(t) - v_2(t)) dt \right|$$

which is clearly not equal to area between the two graphs.

• @David I think you are correct, rather distance travelled by single particle is $\int |v| dt$. – samjoe Nov 30 '17 at 17:07