What is the difference between numerical integration and Riemann integration? I am learning about Riemann integral and i have a question that actually what is the difference between Riemann integration and numerical integration. That is why we should learn Riemann integral though we have numerical integration?
 A: This is a false dichotomy. Riemann integration and numerical integration are not two different methods for calculating an integral, Riemann integration is definition of an integral, and numerical integration is how you calculate one. If you didn't have Riemann integration, numerical integration couldn't exist, since you wouldn't know what you were trying to calculate.
Your question is like asking "If I have an algorithm to compute the digits of $\pi$, why do I need to know the definition of $\pi$?".
The difference between a definition and a method of calculation is seen clearly when you ask the question: how can we tell if a numerical integration algorithm is producing the correct values? For that, you need a definition that tells you what the correct values are. If I write a program that multiplies numbers together, and it says that $5\times 3$ is $7$, how do you know it's wrong? Because there is a definition of multiplication separate from any method for actually doing it, namely, that $5\times 3$ means the amount of objects in $5$ groups of $3$.
A: The Riemann integral is exact (because a limit is used), numerical integration is merely an approximation (because only a finite number of points is used).
A: There's a deeper question here:

Why should we learn Riemann integral

And this question has answers ranging from totally practical to totally theoretical.

Most practical reason:
You use a numerical integration to estimate the correct value of the integral, and the correct value can be calculated using Riemann integration. So basically, numerical integration is the path to a goal, and it makes sense to actually know where you are going (i.e., know what you are aiming for).
Furthermore, you want to know that the numerical estimates converge to the correct value. And how in the world can you do that if you don't previously learn what the correct value is?

Slightly more theoretic:
You need to understand what integrals are to do more complex mathematical things, like solve differential equations. 

Most theoretic:
We study the Riemann integral because we can. The identities stemming from that part of math are beautiful (for example, the fundamental theorem od analysis) and worth studying for their own sake.
