# 数学代写|微积分代写Calculus代写|The Definite integral

#### Doug I. Jones

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## 数学代写|微积分代写Calculus代写|The Definite integral

In this section we consider the limit of general Riemann sums as the norm of the partitions of a closed interval $[a, b]$ approaches zero. This limiting process leads us to the definition of the definite integral of a function over a closed interval $[a, b]$.
Definition of the Definite Integral
The definition of the definite integral is based on the fact that for some functions, as the norm of the partitions of $[a, b]$ approaches zero, the values of the corresponding Riemann sums approach a limiting value $J$. We introduce the symbol $\varepsilon$ as a small positive number that specifies how close to $J$ the Riemann sum must be, and the symbol $\delta$ as a second small positive number that specifies how small the norm of a partition must be in order for convergence to happen. We now define this limit precisely.
DEFINITION Let $f(x)$ be a function defined on a closed interval $[a, b]$. We say that a number $J$ is the definite integral of $\boldsymbol{f}$ over $[\boldsymbol{a}, \boldsymbol{b}]$ and that $J$ is the limit of the Riemann sums $\sum_{k=1}^n f\left(c_k\right) \Delta x_k$ if the following condition is satisfied:
Given any number $\varepsilon>0$ there is a corresponding number $\delta>0$ such that for every partition $P=\left{x_0, x_1, \ldots, x_n\right}$ of $[a, b]$ with $|P|<\delta$ and any choice of $c_k$ in $\left[x_{k-1}, x_k\right]$, we have
$$\left|\sum_{k=1}^n f\left(c_k\right) \Delta x_k-J\right|<\varepsilon .$$
The definition involves a limiting process in which the norm of the partition goes to zero.
We have many choices for a partition $P$ with norm going to zero, and many choices of points $c_k$ for each partition. The definite integral exists when we always get the same limit $J$, no matter what choices are made. When the limit exists we write
$$J=\lim {|P| \rightarrow 0} \sum{k=1}^n f\left(c_k\right) \Delta x_k$$
and we say that the definite integral exists. The limit of any Riemann sum is always taken as the norm of the partitions approaches zero and the number of subintervals goes to infinity, and furthermore the same limit $J$ must be obtained no matter what choices we make for the points $c_k$.

## 数学代写|微积分代写Calculus代写|integrable and Nonintegrable Functions

Not every function defined over a closed interval $[a, b]$ is integrable even if the function is bounded. That is, the Riemann sums for some functions might not converge to the same limiting value, or to any value at all. A full development of exactly which functions defined over $[a, b]$ are integrable requires advanced mathematical analysis, but fortunately most functions that commonly occur in applications are integrable. In particular, every continuous function over $[a, b]$ is integrable over this interval, and so is every function that has no more than a finite number of jump discontinuities on $[a, b]$. (See Figures 1.9 and 1.10. The latter functions are called piecewise-continuous functions, and they are defined in Additional Exercises 11-18 at the end of this chapter.) The following theorem, which is proved in more advanced courses, establishes these results.
THEOREM 1-Integrability of Continuous Functions
If a function $f$ is continuous over the interval $[a, b]$, or if $f$ has at most finitely many jump discontinuities there, then the definite integral $\int_a^b f(x) d x$ exists and $f$ is integrable over $[a, b]$.

The idea behind Theorem 1 for continuous functions is given in Exercises 86 and 87 . Briefly, when $f$ is continuous we can choose each $c_k$ so that $f\left(c_k\right)$ gives the maximum value of $f$ on the subinterval $\left[x_{k-1}, x_k\right]$, resulting in an upper sum. Likewise, we can choose $c_k$ to give the minimum value of $f$ on $\left[x_{k-1}, x_k\right]$ to obtain a lower sum. The upper and lower sums can be shown to converge to the same limiting value as the norm of the partition $P$ tends to zero. Moreover, every Riemann sum is trapped between the values of the upper and lower sums, so every Riemann sum converges to the same limit as well. Therefore, the number $J$ in the definition of the definite integral exists, and the continuous function $f$ is integrable over $[a, b]$.

For integrability to fail, a function needs to be sufficiently discontinuous that the region between its graph and the $x$-axis cannot be approximated well by increasingly thin rectangles. Our first example shows a function that is not integrable over a closed interval.

# 微积分代考

## 数学代写|微积分代写Calculus代写|The Definite integral

$$\left|\sum_{k=1}^n f\left(c_k\right) \Delta x_k-J\right|<\varepsilon .$$

$$J=\lim {|P| \rightarrow 0} \sum{k=1}^n f\left(c_k\right) \Delta x_k$$

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