Here is an easy warm-up from your first year of physics class:

*A car moves at a constant velocity of 6 m/s for 5 seconds. How far does it travel during that time?*

When velocity is constant, you can multiply the velocity by the time and get the displacement:

But it is interesting to look at a velocity vs. time graph that models this situation:

The area between the graph and the t-axis forms a rectangle. So when you are multiplying the velocity (6 m/s) by the time (5 s) you are also finding the area of that rectangle!

But is this *really* an area? Or is it more of an “area”? Look at the lengths of the sides and note the units. We expect the lengths of the sides of a rectangle to be measured in units of distance. But this one has one side measured in velocity units (m/s) and the other in units of time (s). And when you multiply them, you get a unit of distance, not a typical unit of area. So the “area” *on the graph* represents some other physical quantity we care about.

Now here’s something your physics teacher might ask THIS year:

*A car’s velocity increases with time according to the function v(t)=t^{2}. How far does the car travel in the time from t = 0 to t = 5 s ? *

Since we know how to find displacement when the velocity is constant, let’s adapt the idea we used in math class to generate an approximate answer.

Here is our plan:

- Divide the time into intervals, each 1 second long.
- Use the formula
to find the velocities at the beginning of these intervals.*v(t)=t*^{2} - Use those velocities to find the displacements for each one-second interval and then add them up to find our approximate answer.

So you are pretending that instead of gradually increasing, the velocity stayed constant for 1 second at a time, jumping to the next value at the beginning of each second (depending on whether we want to over-estimate or under estimate our answer).

Why don’t we graph what thAT looks like on a velocity vs time graph?

*“It’s like Déjà vu all over again” – Yogi Berra*

I hope this looks familiar! It’s just like what we saw in the last post. Treating the motion as if if were a series of constant-velocity intervals is just like approximating the area under a curved boundary by using a bunch of rectangles!

I used a spreadsheet and put the results of these calculations in the chart below. Take a look at the overestimate and the underestimate. You might recognize those numbers…but once you understand what we are doing here, you won’t have to do these calculations at all!

So now we can say that the cart travels somewhere between 30 and 55 meters. You can probably guess what your teacher will say:

“I like your method, but I wish it were more precise. Can you divide the interval into more, finer intervals? Like maybe 10 of them? Or even 50?

[It’s amazing how your math teacher and physics teacher are so in tune. It’s almost as if they were one person…]

Here’s what 50 rectangles look like (again). Seriously, the only difference is that I labeled the x and y axes to be velocity and time.

And if we want the exact answer, we evaluate the limit:

Except we *still* don’t have to do it this way! We
can still use the Magic Theorem! Really, all that has changed is that we are
calling the function v(t) instead of f(x):

It is true that we have just
found an area under a graph. But what we
also found, *and what we actually wanted
to find*, is the answer to a physics question: how far did the car
move? It is just a helpful connection
that we can think of that as an area.

*IT’S NOT JUST FOR DISPLACEMENT *

Here are some other things we will calculate this year. In each case, we can think of it as an area and then use the Magic Theorem.

1. Impulse exerted by a force that varies with time:

This is also the area under the Force vs time graph.

2. Charge delivered by a current that varies with time:

This is also the area under the current vs time graph.

3. Work done by a force that varies with position:

This is also the area under the force vs. position graph.

Once you are comfortable with this, you can find these things without even drawing the graphs! Integrating will come to mean: adding up infinitely many tiny contributions to an overall total. And as you will see in the next (and last!) post in this series, that’s another powerful way use integration in physics.