Trapezoid rule using MATLAB

Determine the value of the following integral.

In MATLAB create a M-file f.m.

function y=f(x)
y = exp(-x).*sin(x);

The .* is used because this function will be evaluated for a vector, x, so that exp(­x) and sin(x) form vectors. Then what is wanted is the elements of exp(­x) and sin(x) multiplying each other, to give the element in the vector y. This is what the .* does.

Then create a vector of x values:

»x=0:.1:1

0.

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Then create a vector of y values.

»y = f(x)

0.00000

0.09033

0.16266

0.21893

0.26103

0.29079

0.30988

0.31991

0.32233

0.31848

0.30956

Check that the y values are correct by calculating one of them. Finally calculate the area.

»area = trapz(x,y)

area = 0.24491148225216

This is the answer if only 11 points are used. Increase the number to 101, 201, and 401.

»x=0:.01:1;

»y=f(x);

»area = trapz(x,y)

area = 0.24582775039918

»x = 0:.005:1;

»y=f(x);

»area = trapz(x,y)

area = 0.24583469284741

»x = 0:.0025:1;

»y=f(x);

»area = trapz(x,y)

area = 0.24583642846187

These values can be used in a Romberg exptrapolation, as shown in the Table.

Table. Values of Integral using Trapezoid Rule

n	area
101	0.24582775039918
201	0.24583469284741
401	0.24583642846187

Taking the two answers obtained with the smallest x gives

I = 4*(0.24583642846187 ­ 0.24583469284741) / 3 = 0.24583700700002

If all values are used we get:

0.24582775039918	0.24583469284741	0.24583642846187
	0.24583700699682	0.24583700700002
		0.24583700700109

These extrapolated values are all accurate enough, since they achieve 9 digit accuracy even though the original values had at most 5 digit accuracy.