Understanding Calculus

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  Table of Contents

  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 11.1 - Understanding Integration

In the previous chapter we took a highly theoretical look at understanding the close relationship between the integral and its derivative. The process of integration is an infinite summation of the product of a function of x, f(x) , and an infinitely small Δx. This infinite sum from x = a to x = b is the total change in the anti-derivative or integral of its function. The important point to understand was that a small change in a function df is given by its derivative multiplied by an infinitesimal dx.

The goal of this chapter is to understand how to apply integration whenever a variable is changing or a function is undergoing some change over an interval. Consider a mathematical function of the form.

Assuming f and x represent two different conditions, then the situation F is dependent on their product. In other words f and x are two constants whose product defines the dependent dimension, F.

F is therefore, a function of two independent conditions, f and x. If we let x be a variable and f be a constant; we have:

As x changes, F also changes, since its value is only dependent on x and the constant f. Thus:

This should be rather obvious to you. Now what if the condition, f, is also a function of x? In other words, the value of f changes as x changes and is therefore, not a constant over an interval from x = a to x = b. This modifies the relationship to:

Since F is a function of f and x and f is a function of x, then F is also some function of x only, such that;

As the condition, f, is a function of x, then its value changes as x goes from a to b. Therefore can be assumed to be constant only over some infinitely small interval, dx. A small change in df, is proportional to the value of multiplied by the infinitely small change in x, dx.

Dividing both sides by dx:

This extremely important result defines the instantaneous rate change of with respect to x to be equal to . The relationship of to is the same as is to i.e. a function and its derivative. The only difference is the change in notation from . Multiplying both sides by dx again:

Integrating both sides results in:

The result is still a bit abstract and may not seem to be anything new to you. The important point to understand is that F is a function of two independent conditions, f and x. If f is also a function of x, then as x changes, f also changes such that its value is not constant over an interval from x = a to x = b. Therefore an instantaneous change in F, df, needs to be defined over an infinitely small interval over which f(x) is assumed to be constant.

Integrating from some value x = a to x = b gives us the net change in over the interval. Remember that is defined as the anti-derivative of

The result is the fundamental theory behind integral Calculus. Take some time to think deeply and abstractly about what is going on in this derivation. Mathematically it represents the same theory derived in the previous chapter, but conceptually it explains integration in a new way. It allows us to analyze a changing situation in terms of infinitesimal changes which we can sum up to find the net change in the system as one variable changes.

Next section -> Section 11.2 - Geometric Applications

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