Numerical Approximations of Definite Integrals

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Numerical Approximations of Definite Integrals

• Riemann Sums
• Numerical Approximation of Definite Integrals
• Formulae for Approximations
• Properties of Approximations
• Example

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Riemann Sums
A Riemann sum for the integral of a function f
over the interval a,b is obtained by first
dividing the interval a,b into subintervals and
then placing a rectangle, as shown below, over
each subinterval. The corresponding Riemann sum
is the combined area of the green rectangles. The
height of the rectangle over some given
subinterval is the value of the function f at
some point of the subinterval. This point can be
chosen freely.
Taking more division points or subintervals in
the Riemann sums, the approximation of the area
of the domain under the graph of f becomes
better.
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Definite Integrals
Definitions
This definition assumes that the limit does not
depend on the various choices in the definition
of the Riemann sums. This is true if f is
continuous on a,b.
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Numerical Approximations of Definite Integrals
In 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 xj can be freely
chosen according to any rule from the intervals
xj -1 , xj a (i -1)(b - a)/n, a i (b -
a)/n . In left rule approximations, xj
xj-1. In mid rule approximations, xj (xj-1
xj)/2. In right rule approximations, xj xj.
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Formulae for Approximations
Definitions
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Trapezoidal approximations and Simpsons Formula
Depending on the shape of the function in
question, the following approximations are often
better approximations than the previously defined
approximations
Definitions
Trapezoidal Approximation
TRAP(n) (LEFT(n)RIGHT(n))/2
Simpsons Approximation
SIMPSON(n)(2MID(n)TRAP(n))/3.
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Properties of Approximations
Property
This property follows directly from the
definitions as illustrated in the above pictures.
For an increasing positive function every LEFT(n)
is the combined area of rectangles included in
the area of the domain bounded by the graph of
f. Hence we have the first inequality which also
holds if the values of f are not always
positive.
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.
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Properties of Approximations
Error Estimates
This follows directly from the definitions.
If f is decreasing the above inequalities have
to be reversed.
These estimates show that the approximations can
be made as precise as needed simply by increasing
the number n of subintervals.
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Properties of Midpoint Approximations
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.
Property
The approximations MID(n) give the combined
area of rectangles like the one in the picture on
the right. The area of the blue rectangle is the
same as the area of the blue triangle on the
right. Since the triangle is included in the
area bounded by the graph of f, the first
inequality of the above property follows.
The second inequality follows from the fact that
if the graph of a function is concave up, then
the graph lies below a secant line between the
points of intersection.
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Example 1/3
Problem
Solution
The function to be integrated is decreasing in
the interval of integration. Hence we have
for any n. For n 10, we get (compute
using Maple)
The errors of RIGHT(n) or of LEFT(n) estimates
are lt 0.1729.
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Example 2/3
Problem
Solution
In this case the Trapezoidal Approximation and
Simpsons Approximation give better estimates.
To be able to apply these estimates correctly we
first need to figure out the direction of
concavity of the graph of the integrand.
By looking at the sign of the second derivative
we conclude that the graph of the integrand is
concave down in the interval -1,1 and concave
up otherwise.
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Example 3/3
Problem
Solution
Since the graph of the integrand is concave down
in the interval -1,1 and concave up otherwise
we have
TRAP(n, 1,2) here means that we apply the
trapezoidal method in the interval 1,2. The
other terms have similar meanings.
Computing again with Maple for n 10 we get