Re: How Integral Calculus Models Reality
reactor,
Got it. We convert speed to feet per second then we take the shape and use the proper math to find its area using the units of feet per second and time. First was a perfect rectangle so its the lenght times the width the second was a triangle so it is 1/2 the base times the hight. Then we take both areas and add them together to find the 660 feet that the car represented by the blue line as traveled in the same amount of time as compared to the car represented by the red line has traveled which was 293.33 feet. I hope this is correct.
Yes. And just a side note: Making sure our units are the same (i.e. changing the velocity to feet/second so it matches with the seconds of elapsed time) is really not important to the concept of the integral. It is one of those things you just always do in engineering (make sure your units are consistent so they will cancel properly). What we wish to focus on is the concept of the integral calculus as being concerned with the area under a curve. And I think you got that.
So now let us look at what happens when the curves we are interested in become much more irregular. We will see with an irregular velocity curve it becomes harder to compute the precise value of the integral...
Now we see a red line that represents a continuously-varying velocity of a car over the same 10 seconds. So no longer can we use simple shapes like rectangles and triangles to figure out the area under the curve. But we can approximate the area under the curve using a summation technique. Let us "cut up time" into equally-sized strips of (approximately) 0.333 seconds each (and let's call this width "dt"). And we extend each of these uniform-width strips until they touch the wiggly red velocity curve. If we continued to fit these strips under the curve all the way out to 10 seconds, we could then approximate the area under the curve by computing the area of each, individual strip, and then summing up all these stripwise areas from time=0 to time=10. If we wanted a better approximation of the area under the curve, we could simply do the same thing but make the size of the "dt" strip even smaller... say, .1 seconds wide. To get an even better approximation we make "dt" even smaller, say .01 seconds wide. As we make "dt" strips more narrow, obviously there will be more and more strips to add up, but the errors we can see at the top of the strips where they do not precisely fit the red curve become smaller.
This is where the concept of the integral of Newton's calculus comes from. The precise definition of the integral of a function (any function, even curvy ones) is by taking the limit as "dt" approaches zero. The concept of asymtotic limits is an entire area of mathematics we could go into, but we really do not need to in order to understand why the integral is important. For the purpose of dynamic simulations and "6 Degree of Freedom" simulations, we only need to understand that some dynamic variables are the integrals of other variables. Computers can perform numerical integration of these signals to sufficient accuracy by applying the "small strips of dt" model.
So in these two charts we have used to understand the integral, we have come to understand that "The position of a moving body is the numerical integral of the body's velocity." In the form of an equation we would write it as:
Position = Integral(Velocity*dt)
This is the opposite of the definition of the derivative, where we would say:
Velocity = diff(Position)/dt
So the derivative and the integral are inverse functions. Now let's make some other true statements about derivatives and integrals with respect to another measure of motion, a body's acceleration. We can write complementary equations like the above, but with respect to a body's velocity and acceleration:
Acceleration = diff(Velocity)/dt (read as: "Acceleration is the time-derivative of Velocity")
and alternately:
Velocity = Integral(Acceleration*dt) (read as: "Velocity is the time-integral of acceleration")
There is a device that can directly measure a body's accleration. And it is no coincidence that we call such a device an "accelerometer". When we use an accelerometer in an airplane and measure the vertical acceleration of the airplane as it moves, we call this a "g-meter" because it will tell us how many "g's" of acceleration a pilot will feel when he maneuvers the airplane.
Now, if we attach a digital processor to the output of the accelerometer, we can program the processor to numerically integrate the acceleration signal as time ticks by... this numerical integration will yield a measure of the airplane's vertical velocity:
Aircraft Vertical Velocity = Integral(Vertical Acceleration*dt)
And then if we program the processor to take this computed vertical velocity and numerically integrate it once more, it will yield the time-varying position of the body:
Aircraft Vertical Position = Integral(Vertical Velocity*dt)
What I have just described is the basic design of what is called an Inertial Navigation System (INS) used on all aircraft. Each INS has 3 distinct accelerometers mounted at right angles to each other. One points in the vertical direction of the airplane, one points in the lateral direction of the airplane (along its wingspan), and the third points along the nose of the airplane. If we integrate the outputs of all 3 of these accelerometers, as described above, we can constantly compute NINE different dynamic quantities that describe the airplane's linear motion:
Accelerations along the x (fwd/back) , y (left/right), and z (up/down) axes
Velocities along the x, y, and z axes
Positions along the x, y, and z axes
These linear dynamic quantities in three directions (x, y, z) describe THREE of the SIX Degrees Of Freedom (DOFs) of an airplane. The other 3 DOFs are the three rotations the airplane can exhibit by rotating ABOUT the x, y, and z axes. We call these rotations PITCH (rotating about the y axis), ROLL (rotating about the x axis), and YAW (rotating about the z axis).
I will leave it here and see if you have any questions again. If you are good with the above, I can describe the device that an INS uses to measure and calculate the 3 rotational degrees of freedom (DOFs).
RMT
