Jeff Watts

There’s no reason to be scared of math.  Too many people get in a habit of saying things like “Well, I’m just not a math person”, and it becomes a hinderance to them understanding more math than they already know.  This post is about calculus, one of the more abstract mathematics that a lot of people encounter in high school or college.  For many, this is the last math class they take, because it is so abstract that they cannot move further in their math education.

Calculus seems hard for a few reasons:

1. It uses funny symbols that we’re not used to.  (Example:  ∫ )
2. It’s hard to figure out how “normal” people would use it on a day to day basis.
3. There’s all this talk about “finding the area under a curve”, which seems like one of the more useless things one could spend time on.

I’ve taken four or five calculus courses in college and in high school, and every instructor’s opening remark has been something like “Ok, let’s say you need to find the area under a curve.”  That always seems impractical to me, so let me start with a more concrete example.  Let’s say you want to know the distance your car will go when you press the accelerator in different ways.  (Hopefully that seems like a practical application – it would be useful for everything from developing cars with better fuel economy to winning a NASCAR race.)

Example:  you are driving your car at the same speed for a long time.  Let’s say you’re driving 70 miles per hour on a long road where you never have to slow down (think Interstate 10 in West Texas).  Also, let’s say you can drive a long time (say 12 hours) without running out of gas.  If you go the same speed (70 miles per hour) for a certain amount of time (12 hours), then the distance you travel is

70 miles per hour × 12 hours = 840 miles 

If you plotted your car’s speed on a graph, it would look like the medium blue line at the top of the the graph below:

The medium blue line is flat and even though you might not think of it this way, the equation for that line is

y = 70 

This means that your car is always going 70 mph: at the start of the trip, at the end of the trip, and at every point in between – you never change speeds, so y is always equal to 70.

Notice that the way we calculated how far you drove (70 mph × 12 hours = 840 miles) is really just a way of finding the area of the light blue rectangle in the graph above.  The area of a rectangle is just its length times its width – in this case its length is 12 hours (shown by the horizontal, dark blue line) and its width is 70 mph (shown by the vertical, dark blue line).  So the area of the rectangle – is 840 miles.  We just found the area under a “curve”, only our curve was a straight, flat line.  So the area under the “curve” is the same as the total distance your car traveled – 840 miles.

The next post on this topic will show how to do the same thing if your car is speeding up, not just going the same speed all the time.