Lecture in Definite Integrals

(Last Updated On: March 2, 2019)

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.

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 =  

If you have some clarifications. Let me know.

credit: Renato E. Apa-ap, et al.