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We will do it here, too, but in important statements we will try to write things properly.
Since the area is always between the upper and lower sum, equality between the two sums means that we determined A.
This manipulation has the form of taking narrower rectangles.
We have the lower limit a and the upper limit b, giving the integrating interval The variable x is called dummy variable because it is not really important.
Since the Riemann integral is related to the area under the graph of f, the only important information is the shape of the graph.
It seems from the picture that if we made the rectangles really narrow, the error of approximation would be small.