*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.

By taking narrower and narrower rectangles, with a little bit of luck the resulting approximated areas converge to some number, namely the area of the region under the graph of f.We use cookies to offer you a better experience, personalize content, tailor advertising, provide social media features, and better understand the use of our services.To learn more or modify/prevent the use of cookies, see our Cookie Policy and Privacy Policy.takes on both positive and negative values, then the Riemann sum is the sum of the areas of the rectangles that lie above the x-axis minus sum of areas of the rectangles that lie below the x-axis.We use cookies to make interactions with our website easy and meaningful, to better understand the use of our services, and to tailor advertising.The advantage of the upper/lower sum approach is that we do not have to worry about mechanics of this procedure, all the details are hidden in the definition below.Unfortunately, the Riemann approach using rectangles succeeds only if the function f is nice enough, when f is Riemann integrable.This procedure can break down if the approximation errors do not get smaller for narrower rectangles; this depends on the shape of f, for really wild functions the region is strange and it may not make any sense to talk about its area. The widths of rectangles are determined by splitting the interval is split into N segments that determine the sides of the approximating rectangles: Now we have to decide on their heights.There are several methods, here we use the one that is easiest to handle. Their bases are given by the partition, the heights by the supremum or infimum of f in each rectangle.The symbol dx is the differential of x (see for instance Derivatives - Theory - Introduction - Leibniz notation) and here it has only a symbolic role.It is a part of the notation of the Riemann integral, so it is important not to forget it.

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