Riemann Sums and Integral

Riemann sums and Integral was introduced by a German Mathematician Bernhard Riemann.

What are Riemann sums? How do you convert integrals to Riemann sums? How do you calculate Riemann integral? 

Its argument arose when we want to find the area under the curve of a function. Imagine that you are given a rectangle or any other polygon or a function how will you calculate the area under the curve?   

Riemann sums (RS) is there to help you. It is used to approximate the area under the curve of a function by adding shapes like rectangle, trapezoid, etc.

But the question here is how can we make these rectangles?

There are three choices when it comes to making rectangles to approximate the area.

Left Riemann sums:

If we make these rectangles under the curve so that the top-left corner of the rectangles touches the curve as shown in the fig(i) then we call it Left Riemann sums.

fig(i) 

Right Riemann sums:

If we make these rectangles under the curve in a way that the top-right corner of the rectangles touches the curve as shown in the fig(ii) then we call it Right Riemann sums.

fig(ii)

Midpoint Riemann sums:

If we make these rectangles in such a way that the midpoint of the width of the rectangles touches the curve as shown in the fig(iii) then we call it Midpoint Riemann sums.

fig(iii)

So basically, the mechanism of Riemann Integral (RI) is to divide the particular region into non-overlapping rectangles and add the area of those rectangles using Riemann sums. The limit of this Riemann sums gives Riemann integral.

Note:- From now onwards I will be using RI and RS for Riemann integral and Riemann sums respectively.

The smaller the width of rectangles the better the approximation of the area we get which implies that as the width of the rectangles approaches to zero we approach the exact area.

Now, let's see the formal definition of RS to understand the concept nicely.

Riemann Sums:

Let f(x) be a function on a closed and bounded region [a,b]. To find the region S bounded by the interval a and b.

What is the area of region S?

Divide the region S into n non-overlapping subintervals as,

\[\left [ x_{0},x_{1} \right ],\left [ x_{1},x_{2} \right ],...,\left [ x_{n-1},x_{n} \right ]\]

where $a=x_{0}< x_{1}< x_{2}<...< x_{k}<...<x_{n}=b $

This subdivision of interval is known as partition of [a,b] and is denoted by P

The width of these subintervals is represented by $\Delta x$

So, the width of kth subinterval is, $\Delta x_{k}=x_{k}-x_{k-1}$ 

The width of the largest subinterval is represented by ||P|| as norm of the partition P

where, ||B|| is the norm of B and it gives the length of the interval B in n-dimensions.

\[\left \| P \right \|= max\left \{ \Delta x_{1},\Delta x_{2},..., \Delta x_{k},...,\Delta x_{n} \right \}\]

 Make a tagged division in the rectangles $R_{1},R_{2},...,R_{k},...R_{n}$. 

The tags $z_{1},z_{2},...,z_{n}$  can be anywhere in its subinterval, $x_{k-1}\leq z_{k}\leq x_{k} \,\,\,\,\,\,\,k=1,2,...,n$

Then the area of the rectangle $R_k$ is given as, $R_{k}$,

\[A_{k}=f\left(z_{k}\right) \Delta x_{k}\]

So, the area under the curve of a given function can be approximated as,

\[A\approx \sum_{k=1}^{n}A_{k}=\sum_{k=1}^{n}f\left ( z_{k} \right )\Delta x_{k}=f\left ( z_{1} \right )\Delta x_{1}+f\left ( z_{2} \right )\Delta x_{2}+...+f\left ( z_{n} \right )\Delta x_{n}\]

And the number 

$\sum_{k=1}^{n}f\left ( z_{k} \right )\Delta x_{k}$ is the RS.

Can we use RS only on continuous function? 

Yes,we can use RS on a continuous and slightly discontinuous function.

Riemann Integral:

As the limit, $\Delta x$ tends to zero, we approach the exact area under the curve $(A)$, and this limit of the RS is known as RI.

\[A=\lim_{\Delta x\rightarrow 0}\sum_{k=1}^{n}f\left ( z_{k} \right )\Delta x_{k}\]

RI has certain limitations like it can be applied only to bounded functions. So, how can we find the area under the curve of an unbounded region?

One can use Lebesgue integral which allows to integrate unbounded or highly discontinuous function whose Riemann integral does not exist.

You might be thinking is it okay if the width of the rectangles is not the same? Or how can we find area under the curve of a function if the widths of rectangles are different?

So the answer is yes, it is okay because the purpose of the rectangles here is to find the area which we can find even if widths of the rectangles are different but rectangles of uniform width simplify the calculations and hence are preferred.

You might be wondering what is the difference between RS and Definite integral?

Definite integrals represent the area under the curve of a function whereas RS helps us in approximating such areas. So basically, RS is used in approximating definite integrals.