微积分网课代修|积分学代写Integral Calculus代考

微积分网课代修|积分学代写Integral Calculus代考|Newton’s Original Integral

Integration as it was originally conceived of by Newton is the process inverse to differentiation. Such a process, he saw, would have numerous applications in geometry and physics.

In modern language we can describe the situation this way. His language was different but this is essentially how he viewed this process.

Definition $1.1$ (Original Newton Integral). Suppose that $f$ is a function defined on an interval $(a, b)$ and that we can find a continuous function $F:[a, b] \rightarrow \mathbb{R}$ so that
$$F^{\prime}(x)=f(x) \text { for every } x \text { with } a<x<b .$$
Then we will say that $F$ is an indefinite integral of $f$ on $[a, b]$ and we will write
$$\int_{a}^{b} f(x) d x=F(b)-F(a)$$
and call the latter the definite integral of $f$ on $[a, b]$.

微积分网课代修|积分学代写Integral Calculus代考|Beyond the original Newton integral

The key technical fact that allows the descriptive definition just given for the Newton integral to work can be expressed as follows:

LEMMA 1.2. If $F$ and $G$ are both continuous functions on an interval $[a, b]$ and if $F^{\prime}(x)=G^{\prime}(x)$ for all $a<x<b$ then
$$F(b)-F(a)=G(b)-G(a) .$$

微积分网课代修|积分学代写Integral Calculus代考|An extension of the Newton integral

DEFinition $1.4$ (Modified Newton Integral). Suppose that $f$ is a function defined on an interval $(a, b)$ except possibly at finitely many points. Suppose that we can find a continuous function $F:[a, b] \rightarrow \mathbb{R}$ so that $F^{\prime}(x)=f(x)$ for every $x$ with $a<x<b$ with perhaps finitely many exceptions. Then we will say that $F$ is an indefinite integral of $f$ on $[a, b]$ and we will write
$$\int_{a}^{b} f(x) d x=F(b)-F(a)$$
and call the latter the definite integral of $f$ on $[a, b]$.
Note that the definition does not require of the function being integrated that it be defined at every point of the interval $(a, b)$. For example we could ask whether the following integral exists:
$$\int_{a}^{b} \frac{1}{\sqrt{|x-1|} \sqrt{|x-2|} \sqrt{|x-3|} \sqrt{|x-4|}} d x .$$
(Warning: don’t try this one yet!) Most older calculus texts would have trouble with this and might insist that the interval $(a, b)$ not include any of the points 1 , 2,3 , or 4 .

Sometimes they would allow this but require that some particular values be preassigned at the exceptional points (even though they would later turn out to be irrelevant).

微积分网课代修|积分学代写Integral Calculus代考|Newton’s Original Integral

$$F^{\prime}(x)=f(x) \text { for every } x \text { with } a<x<b .$$

$$\int_{a}^{b} f(x) d x=F(b)-F(a)$$

微积分网课代修|积分学代写Integral Calculus代考|Beyond the original Newton integral

$$F(b)-F(a)=G(b)-G(a) .$$

微积分网课代修|积分学代写Integral Calculus代考|An extension of the Newton integral

(修正牛顿积分) 。假设 $f$ 是在区间上定义的函数 $(a, b)$ 除了可能在有限的许多 点上。假设我们可以找到一个连续函数 $F:[a, b] \rightarrow \mathbb{R}$ 以便 $F^{\prime}(x)=f(x)$ 对于每个 $x$ 和 $a<x<b$ 可能有很多例外。然后我们会说 $F$ 是一个不定积分 $f$ 上 $[a, b]$ 我们会写
$$\int_{a}^{b} f(x) d x=F(b)-F(a)$$

$$\int_{a}^{b} \frac{1}{\sqrt{|x-1|} \sqrt{|x-2|} \sqrt{|x-3|} \sqrt{|x-4|}} d x .$$
(警告: 不要尝试这个!) 大多数较旧的微积分课本都会遇到这个问题，并且可能会坚 持认为间隔 $(a, b)$ 不包括任何点 $1 、 2,3$ 或 4 。