Riemann Sum

How to find the area under a function by using the Riemann sum: 1 example and its solution.

Definition

Definition

Let's see how to find the area
under the graph of y = f(x).

First, slice the area to n pieces.

For each piece,
draw a rectangle
whose height is the function value: f(x).

Then, as n → ∞,
the sum of the pieces
becomes the area under the graph of y = f(x).

The sum of the pieces is called the Riemann sum.

Example

Example

Solution

First, slice the area to n pieces.
and draw the rectangle pieces like this.

The width of the area is
1 - 0 = 1.

The area is sliced into n pieces.

So the width of each rectangle is
1/n.

The width of each piece is 1/n.

Then the x value of the right side of the kth piece is
k/n.

Draw the kth rectangle piece below.

The width is 1/n.

The height of the rectangle piece is
f(k/n) = (k/n)2 = k2/n2.

The width is 1/n.
The height is k2/n2.

Then the area, Ak, is
Ak = (k2/n2)⋅(1/n).

There are n rectangle pieces.

So the sum of the n pieces, Sn, is
the sum of (k2/n2)⋅(1/n) as k goes from 1 to n.

Sigma Notation

The variable of the summation is k, not n.

So take the denominators, n2 and n,
out from the summation.

The sum of k2 is
[n(n + 1)(2n + 1)]/6.

Sum of Squares: k2

Then [n(n + 1)(2n + 1)]/6n3.

Recall that
as n → ∞,
Sn becomes the area under y = x2: S.

Sn = [n(n + 1)(2n + 1)]/6n3

So S = the limit of [n(n + 1)(2n + 1)]/6n3.

[n(n + 1)(2n + 1)]/6n3 = [2n3 + ...]/6n3

The highest order term of the numerator is 2n3.
The denominator is 6n3.

Both terms are n3.

So, as n → ∞,
the limit is 2/6.

Indeterminate Form

2/6 = 1/3

So the area of the colored region, S, is 1/3.

Riemann Sum to Definite Integral

Formula

Let's get back to the area under y = f(x).

See the kth rectangle piece
(that looks like a line).
The width is ∆x.
The height is f(xk).
Then the area, Ak, is f(xk)∆x.

Then the Riemann sum, S, is
the limit of the sum of f(xk)∆x.

To change a Riemann sum to a definite integral,

Change the limit and the sigma
to ∫ab: the integral from a to b.
(a: lower limit, b: upper limit)

Change f(xk) to f(x).

And change ∆x to dx.

So S = ∫ab f(x) dx.

So the definite integral means
the sum [∫ab] of the rectangle pieces [f(x) dx].

In the next page,
see how to solve a definite integral.