MATHEMATICS EXTENSION 2
4 UNIT MATHEMATICS
TOPIC 4: INTEGRATION
4.2
THE DEFINITE INTEGRAL
Integration is a very valuable technique for calculations that can be expressed in terms
of a summation. An important theorem relates the limit of a
summation to the definite integral. Consider the continuous
and single-valued function f ( x ) defined in the interval
xa  x  xb . The interval from xa to xb can be divided into N
equal subintervals each of length x where x 
x1  xa
x2  xa  x
x3  xa  2x
xb  xa
and
N
xN 1  xb
Then the sum of the rectangles S N corresponds to
N
S N   f  xi  x
i 1
The fundamental theorem of integral calculus states that as the number of
subintervals N approaches infinity
standard integrals
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N
xb
i 1
xa
limit S N  limit  f  xi  x   f ( x ) dx
N 
N 
definite integral
The definite integral corresponds to the area A under the curve represented by the
function y  f ( x ) in the interval xa  x  xb .
A 
xb
xa
f ( x ) dx
where xa is the lower limit of the integration and xb is the upper limit. The function
f ( x ) is called the integrand and A is called the integral.
The definite integral can be expressed as: if there is some function F(x) which is
differentiable in the interval xa  x  xb and has the derivative f ( x ) then the definite
integral of f ( x ) with respect to x over the interval is
F  xb   F  xa   
standard integrals
xb
xa
f ( x ) dx
definite integral
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Example
Evaluate
 x
2
 2 x  1 dx
2
1
indefinite integral
  3x
2
2
1
 2 x  1 dx
F ( x )    3x 2  2 x  1 dx  x 3  x 2  x  C
F (2)  8  4  2  C
F (1)  1  1  1  C
F (2)  F (1)  8  4  2  C   1  1  1  C 
F (2)  F (1)  11
definite integral
A    3x 2  2 x  1 dx
2
1
A   x 3  x 2  x 
2
1
A   2  1    2 2  12    2  1
3
3
A  7  3  1  11
standard integrals
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Properties of definite integrals
 Interchanging the limits of integration changes the sign of the integral

xb
xa
f ( x ) dx   
xa
xb
f ( x ) dx
 The range of integration can be subdivided

xb
xa
xc
xb
xa
xc
f ( x ) dx   f ( x ) dx  
x a  x c  xb
f ( x ) dx
 Integration by substitution x  h(u ) Need to change the limits of integration

xb
xa
ua
f ( x ) dx   g (u ) du
ub
 Integration of even and odd functions
even function f (  x )  f ( x )
odd function
standard integrals
f ( x)   f ( x)

xa

xa
 xa
 xa
f ( x ) dx  2 
xa
0
f ( x ) dx
f ( x ) dx  0
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Example
Find the area of the triangle with sides 3, 4 and 5.
Equation of hypotenuse y   3/ 4 x 0  x  4
Area of triangle equals area under curve
A    3/ 4  x dx   3/ 8  x 2 
4
4
0
0
A6
standard integrals
xb
A   y dx
xa
A   12   base  height    12   4  3  6
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