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bird · #**100** 07-12-2012, 11:51 PM

Nice article regarding area under the curve:

Sport Compact Car Magazine: Technobabble: August 2001

Quote:

Let's take a step back to see why this is. If you plot anything out on a standard x/y graph (that's Cartesian coordinates, for those of you who like old, stuffy, European sounding names for your graphs), the area under that plot will be the sum of whatever your x-axis is times whatever your y-axis is. That doesn't make any sense at all, does it?

It will, if you look at an example. Say you plot speed on the y-axis and time on the x-axis for a car that is going a constant 1 mile per minute (that's 60 mph) for 5 minutes. Even without a graph, it's pretty simple to figure out how far the car traveled. Speed times time equals distance, so the car traveled five miles. Great.

But guess what? The area under that curve (which happens to be a line, in this case) is exactly the same thing. Since the area under this curve is a rectangle, area simply equals length times width, or speed times time. Amazing. This parity remains even when the curve is a strange shape, so if that same car wandered through traffic making an erratic speed vs. time curve, the area under that curve would still be the distance traveled.

OK, so what? Let's look at something a little more exciting than wandering through traffic. How about accelerating? Acceleration multiplied by time gives you speed, so on a chart of acceleration vs. time, the area under the curve will tell you the final speed. Now here's something we can brag about. All this rambling about charts and graphs starts getting relevant when you remember that acceleration is directly proportional to force, and force is simply what torque becomes when your drive wheels hit the road. So, torque vs. time, along with a few constants like gear ratio, tire size and vehicle weight, will give you speed. Ah, ha! Now you see why the area under the torque curve is the most important of all.

This explains why the 150 hp from Volkswagen's turbocharged, five-valve 1.8T seems so much more powerful than the 150 hp from a Neon. The torque curve of the Volkswagen is tall, wide and of bountiful area.

Of course, it should be noted that torque vs. rpm, as you would get off a dyno chart, is not the same as torque vs. time. Because the car will be going faster at high rpm, the engine will spend less time around 7000 rpm than it did, say, around 3000 rpm. And, of course, to get any truly useful data, you need to look at the force at the contact patch after the engine's torque has been multiplied by the various gear ratios and divided by the radius of the wheel and tire. The area under the curve argument is still good for general comparisons, however, and is most often useful when comparing various modifications to the same car, in which case all the questions about rpm vs. time, gearing and tire size become pretty much irrelevant.