From Wikipedia, the free encyclopedia Rectangle method
Rectangle method
In mathematics, specifically in integral calculus, the rec-
tangle method (also called the midpoint or mid-ordinate
rule) computes an approximation to a definite integral,
made by finding the area of a collection of rectangles
whose heights are determined by the values of the func-
tion.
Specifically, the interval (a,b) over which the function
is to be integrated is divided into N equal subintervals of
length h = (b − a) / N. The rectangles are then drawn so
that either their left or right corners, or the middle of
their top line lies on the graph of the function, with bases
running along the x-axis. The approximation to the inte- Midpoint approximation
gral is then calculated by adding up the areas (base mul-
tiplied by height) of the N rectangles, giving the formula:
where h = (b − a) / N and xn = a + nh.
The formula for xn above gives xn for the Top-left corner
approximation.
As N gets larger, this approximation gets more accu-
rate. In fact, this computation is the spirit of the defini- Top-right corner approximation
tion of the Riemann integral and the limit of this approx-
imation as is defined and equal to the inte-
gral of f on (a,b) if this Riemann integral is defined. Note Error
that this is true regardless of which i’ is used, however
the midpoint approximation tends to be more accurate For a function f which is twice differentiable, the approx-
for finite n. imation error in each section (a,a + Δ) of the midpoint
rule decays as the cube of the width of the rectangle. (For
The different rectangle approximations
a derivation based on a Taylor approximation, see Mid-
point method)
for some ξ in (a,a + Δ). Summing this, the approximation
error for n intervals with width Δ is less than or equal to
n = 1,2,3,...
where n + 1 is the number of nodes
Top-left corner approximation
in terms of the total interval, we know that nΔ = b − a so
we can rewrite the expression:
1
From Wikipedia, the free encyclopedia Rectangle method
for some ξ in (a,b). int i;
Matlab Example Program interval=(b-a)/subintervals;
result=0;
for(i=1;i
Do J=a,b,-.00000001
2
#include
c1=J
Y=F(((c1+c2)/2))
double f(double x){
Mult=Y*.00000001
return sin(x);
Sum=sum+mult
}
c2=c1
End Do
double rectangle_integrate(double a, double b, int subintervals, double (*function)(double
double result;
Print*,Sum
double interval;
2
From Wikipedia, the free encyclopedia Rectangle method
3 Format (F20.5) Return
End End
Double Precision Function f(x)
Double Precision x
See also
• Midpoint method for solving ordinary differential
F=(4)/((x**2)+1) equations
• Trapezoidal rule
• Simpson’s rule
Retrieved from "http://en.wikipedia.org/wiki/Rectangle_method"
Categories: Integral calculus, Numerical integration (quadrature)
This page was last modified on 21 May 2011 at 17:57. Text is available under the Creative Commons Attribution-
ShareAlike License; additional terms may apply. See Terms of use for details. Wikipedia® is a registered trademark of
the Wikimedia Foundation, Inc., a non-profit organization.Contact us
Privacy policy About Wikipedia Disclaimers Mobile view
3