# Riemann Integral

How to find the given area by using the Riemann integral: how to do, example, and its solution.

## How to Do

Let's find the area of the colored region *S*.

First, slice the region vertically to *n* parts.

And draw a rectangle for each slice.

The width of each slice is (*b* - *a*)/*n*.

For the *k*th slice,

the *x* value of the right side is (*b* - *a*)*k*/*n*.

See the *k*th slice.

The width is (*b* - *a*)/*n*.

And the height is *f*( [(*b* - *a*)*k*]/*n* ).

So the area of the *k*th slice, *A*_{k}, is*f*( [(*b* - *a*)*k*]/*n* )⋅(*b* - *a*)/*n*.

Then the sum of the slices, *S*_{n},

is the sum of *f*( [(*b* - *a*)*k*]/*n* )⋅(*b* - *a*)/*n*

as *k* goes from 1 to *n*.

This summation is the Riemann sum.

Sigma notation

As *n* → ∞,

each slice will become so slim

that each slice will fit into the colored region.

So the sum of these slices

is the area of the colored region: *S*.

So *S* is the limit of *S*_{n} as *n* → ∞.

The Riemann sum with the limit

is the Riemann integral.

So, to find the Riemann integral:

1. Slice the region into *n* parts,

and find *A*_{k} (area of the *k*th slice).

2. Write *S*_{n} (Riemann sum).

3. And find *S*, the limit of *S*_{n}.

## Example

Slice the region vertically to *n* parts.

And draw a rectangle for each slice.

The width of each slice is, (1 - 0)/*n*, 1/*n*.

For the *k*th slice,

the *x* value of the right side is (1 - 0)*k*/*n*, *k*/*n*.

See the *k*th slice.

The width is 1/*n*.

And the height is *f*(*k*/*n*) = (*k*/*n*)^{2}.

So the area of the *k*th slice, *A*_{k}, is

(*k*/*n*)^{2}⋅1/*n*.

Then the Riemann sum, *S*_{n},

is the sum of (*k*/*n*)^{2}⋅1/*n*

as *k* goes from 1 to *n*.

It is *k* that goes from 1 to *n*, not *n*.

So take the denominator *n*^{3}

out from the sigma.

As *n* → ∞,

you get the Riemann integral, *S*.

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

Indeterminate form (Part 1)