Proving that definite integral gives the area under the curve using the following definition.

Question

Using this definition how do we prove that the area under the curve is given by the definite integral. The definition of integral I am using is the Riemann's definition.

The definition which uses partitions and tags.

Answer 1

We are speaking of a non-negative function $f$ defined on $[a,b]$ of course.

If $f$ is Riemann integrable, you can find upper and lower sums arbitrarily close to each other.

If you remember the geometrical meaning of upper and lower sums, you are done.