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7 Definite Integrals

Page history last edited by CAI, Sheng 11 years, 10 months ago

Definite Integrals
Bridge Course, August 2012


Intuition

A definite integral is an integral

                                                           Formula

with a upper limit b and a lower limit a. Let f(x) be a function defined on an closed interval [a; b] on real line. The definite integral is defined to be the net signed area of the region bounded by the curve f, the x-axis, the vertical lines x = a and x = b.

Figure 1: Area bounded by y = sin x, x-axis, x = a, x = b


Definition
The fundamental theorem of calculus allows definite integrals to be computed in terms of indefinite inte-grals, since if F is the indefinite integral for a continuous function , then

                                                       Formula

 

Example                                                                                  MAXIMA CODE

Evaluate Formula                                                                                        integrate(x, x, 0, 2)

 

                                        Formula

Note that: The constant of integration does not affect the value of the definite integral.

 

Properties

 

Let f and g be two functions integrable on [a, b]. Assume a<b, then we have

  1. Formula 
  2. Formula 
  3. Formula 
  4. Formula
  5. Formula 
  6. Formula, where the variables FormulaFormula and Formula are called dummy variables
  7.  For definite integrals, integration by parts is defined as follows

       Formula

 

Examples
1. Find the area bounded by y = sin 2x, x-axis, and Formula and Formula
Area of the region:                                                                            integrate(sin(2* x), x,-%pi,%pi)

               Formula

 

Note that: If f is an odd function (i.e. f(-x) = -f(x) for all real number x), Formula for odd functions.

 

2. Find the area bounded by Formula, x-axis, and x = 2 and x = -2
By the definition of absolute value,

                                                            Formula

Therefore area of the bounded region:                     integrate(-3*x, x,-2, 0) + integrate(3*x, x, 0, 2)

 

                              Formula

 

Note that: If f is an even function (i.e. f(-x) = f(x) for all real number x),Formula for even functions.


3. Find the area bounded by Formula,x-axis, and x = 2 and x = 0
By the definition of absolute value,

                                                  Formula

Therefore area of the bounded region:

                              Formula

integrate(%e ^ (1 - x), x, 0, 1) + integrate(%e ^ (-(1 - x)), x, 1, 2)

 

4. By using substitution u = 1 - x, evaluate Formula

Let u=1-x. Then,Formula and Formula and Formula

                              Formula

integrate((1 - x) ^ 5, x,-1, 1)

 

Exercises

  1. Evaluate Formula                     integrate(cos(x)*sin(x), x, -%pi/2, %pi/2) )
  2. Evaluate Formula                                                                         find it yourself
  3. By using substitution Formula, evaluate Formula                        find it yourself
  4. Find the area bounded by Formula, x-axis, x=-10 and x=10                        find it yourself
  5. Find the area bounded by y=|x|, x-axis, x=-10 and x=10                         find it yourself
  6. Find the area bounded by Formula, x-axis, x=-1 and x=3           find it yourself
  7. Find the area bounded by Formula, x-axis, x=0 and x=4              find it yourself

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