Lecture in Definite Integrals

Definite integrals are used to find the area between the graph of a function and the x-axis. If a continuous function f(x) is positive over the domain , then the area under its graph is

Where:

• F(x) is the integral of f(x);


• F(b) is the value of the integral at the upper limit, x=b; and


• F(a) is the value of the integral at the lower limit, x=a.

Note: It does not involve a constant of integration (arbitrary constant) and it gives us a definite value (a number) at the end of the calculation.

Properties of the Definite Integral:

• Integral of a constant:
  
  (a)
  
  (b)
• Linearity:
  
  (a)
  
  (b) if c is a constant.
• Interval Additivity
  
  (a)
  
  (b)
  
  (c) If a < b then it is convenient to define
• Comparison:
  
   If 0 < f(x) < g(x) for all x in [a, b],

The Evaluation Theorem

If f  is a continuous function and F is an antiderivative of f,   F'(x) = f(x), then

 

 Example: Using the Theorem, Find the value of  .

• Applying what you learned, an antiderivative of x² is
• Then, evaluate and substitute the limit. 

= with limit 0 to 1 =  

At this point, you are ready to answer some problems involving definite integral. Follow the link to start. Definite Integral - Set 1 Problems

If you have some clarifications. Let me know.

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