Why do we use Riemann approximations when we can find actual area by using integrals

Question

I am a calculus 1 student. I was wondering that if Riemann sums only give us an approximation(either over-estimate or under-estimate) the area under the curve, Why do we celebrate Riemann sums(considering it came out in the 19th century) when we can actually find the exact area using integrals.

Answer 1

The Riemann sums are used to construct the integral, to define the object. When the functions to be integrated are "nice enough" you have learned a simple formula to compute the integral (involving primitives), but this rule does not define the integral, nor does it allow to compute every integral.