Understanding Calculus

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  Preface
  1. Why Study
  Calculus
  2. Numbers
  3. Functions
  4. The Derivative
  5. Differentiation
  6. Applications
  7. Free Falling
  Motion
  8. Understanding
  Derivative
  9. Derivative
  Approximations
  10. Integration
  Theory
  11. Understanding
  Integration
  12. Differentials

  Inverse Functions
  Exponents
  Exponential
  Functions
  Applications of
  Exponential
  Functions
  Sine and Cosine
  Function
  Sine Function
  Sine Function -
  Differentiation and
  Integration
  Oscillatory Motion
  Mean Value
  Theorem
  Taylor Series
  More Taylor Series
  Integration
  Techniques

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CHAPTER 11

Chapter 11 - Understanding Integration

Section 11.3 - The Systematic Approach to Integration

Integration is a step-by-step process used to analyze functions that are changing. While studying the geometric applications, it was often easier to simply look at the graph and identify a differential element that corresponds to a physical situation such as an area or volume. Such a direct approach is generally not recommended since most applications of integration are entirely independent of any geometry. Furthermore the direct approach does not reflect any logical method or orderly analysis. For these reasons we shall study a systematic approach to integration that is applicable to any changing situation.

By letting x and f represent something other than mere abstractions we will need to integrate our mathematical functions in a more systematic and obvious method. As we shall see, setting up integrals is often complicated by the presence of other dimensions in the equations such that success is determined by realizing which are constants and which are variables. The best way to get a proper grasp of what is going on in the following series of examples is not to feel overwhelmed by the apparent complexity of the situation.

We will first present a four step procedure for setting up an integral and evaluating it. This is called the systematic approach to integration and is derived from our understanding of integration as summation process involving one changing dimension.

Step 1- Determine the form of the functional relationship between the interacting conditions. This is synonymous with writing the equation, where every dimension is assumed to be a constant:

Step 2- Identify which dimension is changing with respect to another dimension and determine the independent variable. If f were given as a function of x, then its value would change as x changed. Thus x is our independent variable.

Step 3- Write the differential dF, as a product of f(x) and an infinitely small change in the independent variable x, dx.

Step 4- Integrate both sides of the function from some value x=a to x=b to calculate the net change in the dependent dimension F.

To understand how to apply the systematic approach, Consider the problem of finding the work done by a gravitational field on a space shuttle. Jules Verne, the master of science fiction, wrote in his popular book Round the Moon about a space shuttle that was launched into orbit by literally firing it from a large cannon with a long muzzle. The ship veered of course and instead of landing on the moon, it flew towards the side of the moon, around the dark-side or back of the moon then came out the other side, directed towards earth. Luckily it did find its way back to earth safely, by landing in the sea. Enough said of that, the problem we want to consider is how much TNT would be needed to send the shuttle to the moon.

There are essentially two forces, both due to gravity, acting on the shuttle. That of the earth, which attracts the shuttle back towards the earth and that of the moon which attracts it forward. The equation for the gravitational force between two bodies is given by Newton's Law of gravitation is where G is a constant, M the mass, and d the distance between the two bodies. The derivation of this formula is not difficult but what exactly is gravity and what causes it, nobody knows. The fact is that there is an observed force determined by this equation.

We can simplify this problem by finding the location of the point where the gravitational force of the earth is equal and opposite to the moons gravitational force. We do this by setting the two equation equal to each other:

Solving for d gives us 200,000 km from the center from the earth, remember gravitational forces are measured from the centers of the body. Once the shuttle reaches this point, the moons gravity will be stronger than the earths and will pull the shuttle towards it. No extra energy will be required.

We can now state the problem as find the total amount of energy required to send the shuttle to this midway point. Then use this answer to determine how much TNT to use. In our systematic approach. step one is to always write the equation first. In this case we are looking for total energy and energy is given by the simple equation:

Step two is to identify which variable is changing, or which variable changes with respect to another constant. From the given equation for gravitational force between two bodies we had force given as function of distance:

There are two forces acting in opposite directions on the shuttle, that of the earth and that of the moons, where the earth's force will always be greater till the midpoint is reached.

We can now write the equation:

This tells us that as the force is changing with respect to the distance and is not constant over a distance d.

In step three we determine our independent variable and then write our differential over an infinitely small interval. From the above equation it is clear that Energy is entirely dependent on distance which tells us that d is our independent variable, since everything else, including force, is dependent on it.

We can now turn the d after the f(d) into an infinitely small Δd. When doing this we must then convert the Energy also into an infinitely small ΔE. This tells us that the force at a particular distance, multiplied by small distance Δd gives the energy done by the force over that interval or:

In the fourth and final step we determine which values, the dependent variable goes from and ends at. These values become our limits of integration so we can now take the infinite sum of F(d) Δd over an interval to find the total energy done. In our example the distance goes from the radius of the earth, 6400 km to the midpoint, 200,000 km away:

Replacing F(d) with the value we derived it to be gives us:

Evaluating this integrals gives us:

Therefore the shuttle must have at least 100000000 Joules of energy before leaving the earth's atmosphere, by the time it reaches the mid-way point it will have almost no energy left as all was used up by the opposing gravitational pull of the earth and the moon. Given that 1 kg of TNT will give 10 Joules of energy, 10000000 kg of TNT will be needed to blast the shuttle out of the canon.

Next section -> Section 11.4 - Engineering Applications

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