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 xaxis, the vertical lines x = a and x = b.
Figure 1: Area bounded by y = sin x, xaxis, x = a, x = b
Definition
The fundamental theorem of calculus allows definite integrals to be computed in terms of indefinite integrals, 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




 , where the variables , and are called dummy variables
 For definite integrals, integration by parts is defined as follows
Examples
1. Find the area bounded by y = sin 2x, xaxis, 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 , xaxis, 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 ,xaxis, 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=1x. Then, and and
integrate((1  x) ^ 5, x,1, 1)
Exercises
 Evaluate integrate(cos(x)*sin(x), x, %pi/2, %pi/2) )
 Evaluate find it yourself
 By using substitution , evaluate find it yourself
 Find the area bounded by , xaxis, x=10 and x=10 find it yourself
 Find the area bounded by y=x, xaxis, x=10 and x=10 find it yourself
 Find the area bounded by , xaxis, x=1 and x=3 find it yourself
 Find the area bounded by , xaxis, x=0 and x=4 find it yourself
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