Continuous Functions Are Integrable

number of rectangles

23

function

x/2

2

x

/2

3

x

/4

log( x + 1 )

sin(x/5)

This Demonstration illustrates a theorem from calculus: A continuous function on a closed interval

[a,b]

is integrable, which means that the difference between the upper and lower sums approaches 0 as the length of the subintervals approaches 0.