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# 数学代写|微积分代写Calculus代考|MATH2310

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## 数学代写|微积分代写Calculus代考|Infinite Limits of Integration

Consider the infinite region (unbounded on the right) that lies under the curve $y=e^{-x / 2}$ in the first quadrant (Figure 8.13a). You might think this region has infinite area, but we will see that the value is finite. We assign a value to the area in the following way. First find the area $A(b)$ of the portion of the region that is bounded on the right by $x=b$ (Figure $8.13 b)$.
$$\left.A(b)=\int_0^b e^{-x / 2} d x=-2 e^{-x / 2}\right]0^b=-2 e^{-b / 2}+2$$ Then find the limit of $A(b)$ as $b \rightarrow \infty$ $$\lim {b \rightarrow \infty} A(b)=\lim {b \rightarrow \infty}\left(-2 e^{-b / 2}+2\right)=2 .$$ The value we assign to the area under the curve from 0 to $\infty$ is $$\int_0^{\infty} e^{-x / 2} d x=\lim {b \rightarrow \infty} \int_0^b e^{-x / 2} d x=2 .$$

DEFINITION Integrals with infinite limits of integration are improper integrals of Type $I$.

1. If $f(x)$ is continuous on $[a, \infty)$, then
$$\int_a^{\infty} f(x) d x=\lim _{b \rightarrow \infty} \int_a^b f(x) d x .$$
2. If $f(x)$ is continuous on $(-\infty, b]$, then
$$\int_{-\infty}^b f(x) d x=\lim _{a \rightarrow-\infty} \int_a^b f(x) d x .$$
3. If $f(x)$ is continuous on $(-\infty, \infty)$, then
$$\int_{-\infty}^{\infty} f(x) d x=\int_{-\infty}^c f(x) d x+\int_c^{\infty} f(x) d x,$$
where $c$ is any real number.
In each case, if the limit exists and is finite, we say that the improper integral converges and that the limit is the value of the improper integral. If the limit fails to exist, the improper integral diverges.

## 数学代写|微积分代写Calculus代考|Integrands with Vertical Asymptotes

Another type of improper integral arises when the integrand has a vertical asymptote-an infinite discontinuity – at a limit of integration or at some point between the limits of integration. If the integrand $f$ is positive over the interval of integration, we can again interpret the improper integral as the area under the graph of $f$ and above the $x$-axis between the limits of integration.

Consider the region in the first quadrant that lies under the curve $y=1 / \sqrt{x}$ from $x=0$ to $x=1$ (Figure $8.12 \mathrm{~b}$ ). First we find the area of the portion from $a$ to 1 (Figure 8.16):
$$\left.\int_a^1 \frac{d x}{\sqrt{x}}=2 \sqrt{x}\right]a^1=2-2 \sqrt{a} .$$ Then we find the limit of this area as $a \rightarrow 0^{+}$: $$\lim {a \rightarrow 0^{+}} \int_a^1 \frac{d x}{\sqrt{x}}=\lim {a \rightarrow 0}(2-2 \sqrt{a})=2 .$$ Therefore the area under the curve from 0 to 1 is finite and is defined to be $$\int_0^1 \frac{d x}{\sqrt{x}}=\lim {a \rightarrow 0^{+}} \int_a^1 \frac{d x}{\sqrt{x}}=2 .$$

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。