Definite Integrals
Bridge Course, August 2012

Intuition

A definite integral is an integral

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

Example                                                                                  MAXIMA CODE

Evaluate                                                                                         integrate(x, x, 0, 2)

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

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

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

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

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

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

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

Therefore area of the bounded region:

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

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

Let u=1-x. Then, and  and 

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

Exercises

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