Approximation of Integration (The Lower Sums and the Upper Sums)

The sum of the areas of the inscribed rectangles is called a lower sum.

The sum of the areas of the circumscribed rectangles is called an upper sum.

Here is the example.

Find the upper and lower sums for the region bounded by the graph of f(x) = x^2 and the x-axis between x=0 and x=2.

The Lower Sum, using the left endpoints (Figure 2-01)

with(student);

>f := x -> x ;

>leftbox(f(x), x = 0 .. 2);

The Upper Sum, using the right endpoints (Figure 2-02)

with(student);

>f := x -> x ;

>rightbox(f(x), x = 0 .. 2);

 

 

Limits of the Lower and Upper Sums

Let f be continuous and nonnegative on the inverval [a,b]. When n is going to infinity, the limits of both the lower and upper sums exist and are equal to each other.