First, let’s set apart the differences. Numerical integration is just a set of methods or algorithms to find the value of definite integrals. By no means is it possible to find indefinite integrals.
Since integration is just the sum of infinite many rectangles under the curve following numerical integration you can only sum up finite rectangles thus leaving scope for error always though mostly the accuracy under the desired limit could be achieved.
Did you know that the scientific calculators use numerical integration to find the value of definite integrals or perform any such operation?
There are some cases where it is not possible to find the integral of the function or the function under consideration is unknown and the use of numerical integration is justified in these types of situations but the question that comes to our mind, why do we need to approximate derivatives?
Even though the derivative of all the functions can be analytically calculated there are still a few cases where we have to approximate the derivative.
If we know the value at few points but the function itself is unknown and there is also a case where the function might not exist but we are still interested in studying the changes in the point or the exact function just requires a lot of computation power.
Consider at you’re modeling a multi-degree of freedom system and you want to study the nature of its acceleration or velocity. If you know the value of displacement with time at multiple locations, you can use numerical analysis to generate a function that would give you desired results.
Similarly, these methods become quite useful when dealing with real-life problems. The next time you solve a solid structure problem. Try developing code on Matlab for the system and compare your result with the result from the exact equation and make changes in your algorithm to minimize your error.
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