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Empirical Model Building
Published in Adedeji B. Badiru, Data Analytics, 2020
Riemann integral is the limit of the Riemann sums of a continuous function as the partitions get smaller and smaller. This approach is also applicable to functions that are not too seriously discontinuous. The Riemann sum formula provides a precise definition of the definite integral as the limit of an infinite series: ∫abf(x)dx=limn→∞∑i=1nf(xi)(b−an)
Integration Theory
Published in James K. Peterson, Basic Analysis I, 2020
We can also interpret the Riemann sum as an approximation to the area under the curve. The partition (closed circles) is P={1.0,1.5,2.6,3.8,4.3,5.6,6.0}.For the evaluation set (open circles) E={1.1,1.8,3.0,4.1,5.3,5.8}.
Infinite Series
Published in John Srdjan Petrovic, Advanced Calculus, 2020
We have seen that finite Riemann sums can be used to approximate definite integrals. However, they do just that—approximate. In order to obtain the exact result, we need the limits of these sums. Similarly, Taylor polynomials provide approximations of functions, but there is the error term rn. Remark 4.5.3 reminds us that, as n → ∞, this error goes to 0, which means that we would have an exact equality between the function and the Taylor polynomial if the latter had infinitely many terms. Such sums (with infinitely many terms) are the infinite series and they will be the focus of our study in this chapter.
Reasoning about geometric limits
Published in International Journal of Mathematical Education in Science and Technology, 2021
Andrijana Burazin, Ann Kajander, Miroslav Lovric
This strategy of taking ‘snapshots’ can be extended to advanced concepts. As an illustration, we examine the calculation of the area of a region under a curve by using the (approximating) Riemann sums. We take the region to be the triangle under the line y = x and over the interval [0,1]; see Figure 11.