Riemann Integral

Riemann Integral

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

How to Do

To find the given area by using the Riemann integral, first write the sliced area as a Riemann sum, then find the limit of the Riemann sum.

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

See the kth slice.

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

So the area of the kth slice, Ak, is
f( [(b - a)k]/n )⋅(b - a)/n.

Then the sum of the slices, Sn,
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 Sn 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 Ak (area of the kth slice).
2. Write Sn (Riemann sum).
3. And find S, the limit of Sn.

Example

Find the area of the colored region by using the Reimann Integral. The region whose boundaries are y = x^2, x-axis, and x = 1.

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

See the kth slice.

The width is 1/n.
And the height is f(k/n) = (k/n)2.

So the area of the kth slice, Ak, is
(k/n)2⋅1/n.

Then the Riemann sum, Sn,
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 n3
out from the sigma.

Sum of Squares (k2)

As n → ∞,
you get the Riemann integral, S.

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

Indeterminate form (Part 1)