Why is integration used so widely though they are just approximation?

Question

Integration is used so widely in higher areas like rocket science etc. As integration is just approximation, even $0.01$ Pascal pressure error might bring a large disaster right! Even fuel consumption error might also bring a big disaster. So does the approximation in integration don't effect any thing (as no major disasters occur in reality) and can anybody say where I am wrong in given above examples.

Answer 1

This question appears based on a misunderstanding:

integration is just approximation.

Integration is in fact a fully exact technique. It may feel like an approximation since $\int_a^b f(x)dx$ is defined as the limit of a family of approximations,$^*$ but that's not the case: the whole magic of limits is that by aggregating a bunch of approximations in a particular way we can in fact get the exactly correct result.

For example, remember that $0.999999...=1$ even though by definition $0.9999999...=\lim_{n\rightarrow\infty}\sum_{i=1}^n 9\cdot 10^{-i}$ and each partial sum $\sum_{i=1}^n 9\cdot 10^{-i} $ is strictly $<1$. Integration is more complicated than this, but the underlying logic is the same.

Now there are techniques which do yield only approximate results, and which are employed when the actual integral is difficult or impossible to calculate - namely, numerical integration. However, these techniques also provide error bounds (and proofs of their adequacy), so they still let us produce answers satisfying a given accuracy requirement.

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$^*$Note that the characterization of the integral via the fundamental theorem of calculus as $\int_a^bf(x)dx=F(b)-F(a)$ for $F$ an antiderivative of $f$ also involves a limit - namely in the definition of the derivative. We can't escape the role of limits in calculus, which is why it's important to understand that they in fact do provide exact results (despite what the language we use to describe them may suggest).