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.
To be on the safe side, we look at the largest and smallest possible (and reasonable) rectangles, obtaining the upper sum and the lower sum: Note that since the function f is bounded, the suprema and infima in the definition always exist finite. The upper and lower sum is shown in the following pictures.
On the left, the area of the shaded region is the upper sum; on the right, the area of the shaded region is the lower sum.
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.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.So if we decide to use a different variable in the same formula, the shape and therefore the integral stay the same.Thus, for instance, Indeed, the area under the same piece of the given parabola is always the same, regardless of what letter we write next to the horizontal axis.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.The error of approximation is then smaller, which means that the upper sum gets smaller (and therefore closer to A) and the lower sum gets larger (and closer to A).In the next picture, compare the error of approximation of the upper and lower sum when we refine the partition. Fw-300 #ya-qn-sort h2 /* Breadcrumb */ #ya-question-breadcrumb #ya-question-breadcrumb i #ya-question-breadcrumb a #bc .ya-q-full-text, .ya-q-text #ya-question-detail h1 html[lang="zh-Hant-TW"] .ya-q-full-text, html[lang="zh-Hant-TW"] .ya-q-text, html[lang="zh-Hant-HK"] .ya-q-full-text, html[lang="zh-Hant-HK"] .ya-q-text html[lang="zh-Hant-TW"] #ya-question-detail h1, html[lang="zh-Hant-HK"] #ya-question-detail h1 /* Trending Now */ /* Center Rail */ #ya-center-rail .profile-banner-default .ya-ba-title #Stencil . Bgc-lgr .tupwrap .comment-text /* Right Rail */ #Stencil . Fw-300 .qstn-title #ya-trending-questions-show-more, #ya-related-questions-show-more #ya-trending-questions-more, #ya-related-questions-more /* DMROS */ .