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  Integration-Left, Right and Trapezoid Rules

Integration: Left, Right and Trapezoid Rules

The Left and Right endpoint rules:

We wish to estimated a definite integral:

940_endpoint rules.jpg

wheref(x) is a continuous function. In calculus we erudite that integrals are (signed) areas as well as canare approximated by total of smaller areas such as the areas of rectangles. We begin by choosingpoints {xi} that subdivide [a, b]

a = x0 < x1< . . . < xn−1< xn= b.

The subintervals [xi−1, xi] conclude the width Δxi of all of the approximating rectangles. For the height we learned that we can select any height of the function f(x*i) where x*i ∈ [xi−1, xi].

The resultant approximation is:

674_result approximation.jpg

To utilization this to approximate integrals with actual numbers we need to have a specific x*i in each interval. The two simplest as well as worst ways to choose x*i are as the left-hand point or the right-handpoint of each interval. This provides concrete approximations which we denote by Ln and Rn given by:

2212_right handpoint.jpg

function L = myleftsum(x,y)
% produces the left sum from data input.
% Inputs: x -- vector of the x coordinates of the partition
% y -- vector of the corresponding y coordinates
% Output: returns the approximate integral
n = max(size(x)); % safe for column or row vectors
L = 0;
for i = 1:n-1
L = L + y(i)*(x(i+1) - x(i));
End

457_left and right sums.jpg

 
The left and right sums, Ln and Rn.

Habitually we can take {xi} to be evenly spaced with each interval having the same width

h = (b − a)/ n

Where n is the amount of subintervals. If this is the situation then Ln and Rn simplify to:

1255_number of subintervals.jpg

The foolishness of choosing left or else right endpoints is illustrated As you can observe fora very simple function like f(x) = 1+.5x, all rectangle of Ln is too short while each rectangle of Rn is too tall. This will grasp for any increasing function. For decreasing functions Lnwill foreverbe too large while Rnwill always be too small.

The Trapezoid rule:

Knowing that the errors of Lnas well as Rnare of opposite sign a very reasonable way to get a better approximation is to take an average of the two. We will call the fresh approximation Tn:

Tn=  (Ln+ Rn)/ 2

This method as well has a straight-forward geometric interpretation. On every sub rectangle we are using:

Ai= {(f(xi−1) + f(xi))/2}*Δxi

Which is precisely the area of the trapezoid with sides f(xi−1) and f(xi). We therefore call the method the trapezoid method.
We are able to rewrite Tn as:

1650_trapezoid rule.jpg

1914_trapezoid rule.jpg

The trapezoid rule Tn.

In the evenly spaced case we are able to write this as:

Tn= {(b – a)/2n}(f(x0) + 2f(x1) + . . . + 2f(xn−1) + f(xn))

Caution- The convention utilized here is to begin numbering the points at 0 that is x0 = a this permits n to be the number of subintervals and the index of the last point xn. Nevertheless, Mat lab’s indexing convention begins at 1. Therefore when programming in Mat lab the first entry in x will be x0

That is x1= x0 and xn+1= xn.

If we are given data about the function moderately than a formula for the function frequently the data are not evenly spaced. The subsequent function program could then be used.

function T = mytrap(x,y)
% calculates the Trapezoid rule estimate of the integral from input data
% Inputs: x -- vector of the x coordinates of the partitian
% y -- vector of the corresponding y coordinates
% Output: returns the approximate integral
n = max(size(x)); % safe for column or row vectors
T = 0;
for i = 1:n-1
T = T + .5*(y(i)+y(i+1))*(x(i+1) - x(i));
end

Utilizing the Trapezoid rule for areas in the plane:

In multi-variable calculus you were theoretical to learn that you can calculate the area of a region R in the plane by calculating the line integral:

A = −ΦCydx

Where C is a counter-clockwise curve around the boundary of the region. We are able to represent such a curve by consecutive points on it that is x¯= (x0, x1, x2, . . . , xn−1, xn), and y¯= (y0, y1, y2, . . . , yn−1, yn).

Since we are assuming the curve ends where it starts we require (xn, yn) = (x0, y0). Applying the trapezoid technique to the integral gives:

378_trapezoid method.jpg

This formula afterwards is the basis for calculating areas when coordinates of boundary points are known however not necessarily formulas for the boundaries such as in a land survey.

In the following script we can utilize this method to approximate the area of a unit circle using n points on the circle

% Calculates pi utilizing a trapezoid approximation of the unit circle.
format long
n = 10;
t = linspace(0,2*pi,n+1);
x = cos(t);
y = sin(t);
plot(x,y)
A = 0
for i = 1:n
A = A - (y(i)+y(i+1))*(x(i+1)-x(i))/2;
end

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