numerical approximations of definite integrals mika seppälä
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Numerical Approximations of Definite Integrals
Mika Seppälä
Mika Seppälä: Numerical
Integration
Riemann Sums
The definite integral of a positive function f over an interval [a,b]
has been defined by Riemann sums which approximate the area under the graph of f.
Taking more division points in the Riemann sums, the approximation of the area of the domain under the graph of f becomes better.
f( )b
ax dx
Mika Seppälä: Numerical
Integration
Definite Integrals
1
0 1 n-1 n 1
11
1
Let { |j=0,..., } be a decomposition of
the interval [ , ] into subintervals [ , ], i.e.
0 <x < <x <x =b. Let [ , ] .
Riemann sum: S (f) f( )( )
Let | | max{ | 1,2
j
j j
j j j
n
D j j jj
j j
D x n
a b x x
x x x j
x x
D x x j
, , }.n
0Definition. f( ) lim ( ).
b
Da Dx dx S f
This definition assumes that the limit does not depend on the various choices in the definition of the Riemann sums.
Mika Seppälä: Numerical
Integration
Numerical Approximations of Definite IntegralsIn view of the definition of the definite integral
we may approximate its value by choosing the decomposition D to be a decomposition of the interval [a,b] into subintervals of length (b-a)/n for some positive integer n. The points j can be freely chosen according to any rule from the intervals [xj-1,xj].
In left rule approximations, j=xj-1.
In mid rule approximations, j=(xj-1+ xj)/2.
In right rule approximations, j=xj.
f( )b
ax dx
Mika Seppälä: Numerical
Integration
Formulae for Approximations
1 1
Consider a function f on an interval [ , ], .
Let , and be a positive integer.
The following sums approximate f( ) .
LEFT( ) f( ( 1) ) RIGHT( ) f( )
MID( ) f(
b
a
n n
k k
a b a b
b ax n
n
x dx
n a k x x n a k x x
n
1
1( ) )
2
n
k
a k x x
Mika Seppälä: Numerical
Integration
Trapezoidal approximations and Simpson’s Formula
Depending on the shape of the function in question, the following approximations are usually better:
Trapezoidal Approximation: TRAP(n) = (LEFT(n)+RIGHT(n))/2
Simpson’s Approximation: SIMPSON(n)=(2MID(n)+TRAP(n))/3.
Mika Seppälä: Numerical
Integration
Properties of Approximations
If the function f is increasing:
LEFT( ) f( ) and RIGHT( ) f( )b b
a an x dx n x dx
If f is strictly increasing – like in the above picture – then the above inequalities are also strict. If f is decreasing, then the direction of the above inequalities must be changed.
Mika Seppälä: Numerical
Integration
Properties of ApproximationsFor any function f,
LEFT( ) RIGHT( ) ( ) ( ) .
If f is increasing, then
LEFT( ) f( ) RIGHT( ).
If f is decreasing, the directions in the ab
b
a
b an n f a f b
n
n x dx n
ove inequalities are reversed.
This implies that for montone functions (either increasing or decreasing),
LEFT( ) f( ) ( ) ( )
and RIGHT( ) f( )
b
a
a
b an x dx f a f b
n
n x dx
( ) ( ) .b b a
f a f bn
These estimates show that the approximations can be made as precise as needed simply by increasing the number n of subintervals.
Mika Seppälä: Numerical
Integration
Properties of Midpoint Approximations
MID( ) f( )b
an x dx
A function which is concave up has the property that its graph lies above any tangent line. This observation leads to the following estimate valid for functions that are concave up.
The blue triangle on the right has been obtained by letting the top side of the rectangle on the left turn around the point where it intersects the graph of the function f. Since this is also the midpoint of the top side, the areas of the two blue domains are the same.