Trapezium Rule


Exercises for students
Solve the following problems on paper and check your working using the above applet:

Use the trapezium rule to find an approximate value for the definite integral of sqrt(1+x^2) dx from x=0 to x=2, using 4 intervals of equal width.
Repeat with 1 interval, 2 intervals & 9 intervals of equal width.

Use the trapezium rule to find an approximate value for the definite integral of 1/(1 + ln(x)) dx from x=1 to x=2.5 , using 3 intervals of equal width.

Use the trapezium rule to find an approximate value for the definite integral of 1/sqrt(1 + sqrt(x)) dx from x=0 to x=1, using 4 intervals of equal width.

The trapezium rule, with 4 intervals of equal width, is to be used to find an approximate value for ∫₀¹ exp(-x) dx.
Explain, with the aid of a sketch, whether the approximation will be greater or less than the exact value of the integral.
Calculate the approximate value, giving your answer correct to 3 decimal places.

Use the trapezium rule with 5 ordinates to find an approximate value for ∫₀¹ x*exp(x^2) dx.


Summary

The Trapezium Rule approximates the area under the curve by the formula:

[ y₀ + 2 ( y₁ + y₂ + ... + y_n-1 ) + y_n ] * h / 2

where n = number of ordinates
a = first ordinate
b = last ordinate
h = width = (b - a) / n
y_i = f ( a + i * h )



You can define the function using the following operators:

+	-	*	/	^
abs( )	sqrt( )	ln( )	exp( )	pi
	sin( )	cos( )	tan( )
	asin( )	acos( )	atan( )
	sinh( )	cosh( )	tanh( )

Right-click here to download this page, the Java Class File and unzip the Javathings Math Package.
This applet uses the com.javathings.math package developed by:
Patrik Lundin
patrik@javathings.com
http://www.javathings.com