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# 数学代写|微积分代写Calculus代考|Math323 Numerical Methods of Integration

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## 数学代写|微积分代写Calculus代考|Numerical Methods of Integration

While there are many integrals that we can calculate explicitly, there are many others that we cannot. For example, it is impossible to evaluate
$$\int e^{-x^2} d x$$
That is to say, it can be proved mathematically that no closed-form antiderivative can be written down for the function $e^{-x^2}$. Nevertheless, $(*)$ is one of the most important integrals in all of mathematics, for it is the Gaussian probability distribution integral that plays such an important role in statistics and probability.

Thus we need other methods for getting our hands on the value of an integral. One method would be to return to the original definition, that is to the Riemann sums. If we need to know the value of
$$\int_0^1 e^{-x^2} d x$$
then we can approximate this value by a Riemann sum
$$\int_0^1 e^{-x^2} d x \approx e^{-(0.25)^2} \cdot 0.25+e^{-(0.5)^2} \cdot 0.25+e^{-(0.75)^2} \cdot 0.25+e^{-1^2} \cdot 0.25$$

## 数学代写|微积分代写Calculus代考|THE TRAPEZOID RULE

The method of using Riemann sums to approximate an integral is sometimes called “the method of rectangles.” It is adequate, but it does not converge very quickly and it begs more efficient methods. In this subsection we consider the method of approximating by trapezoids.

Let $f$ be a continuous function on an interval $[a, b]$ and consider a partition $\mathcal{P}=\left{x_0, x_1, \ldots, x_k\right}$ of the interval. As usual, we take $x_0=a$ and $x_k=b$. We also assume that the partition is uniform.

In the method of rectangles we consider a sum of the areas of rectangles. Figure $8.44$ shows one rectangle, how it approximates the curve, and what error is made in this particular approximation. The rectangle gives rise to a “triangular” error region (the difference between the true area under the curve and the area of the rectangle). We put quotation marks around the word “triangular” since the region in question is not a true triangle but instead is a sort of curvilinear triangle. If we instead approximate by trapezoids, as in Fig. $8.45$ (which, again, shows just one region), then at least visually the errors seem to be much smaller.

## 数学代写|微积分代写Calculus代考|Numerical Methods of Integration

$$\int e^{-x^2} d x$$

$$\int_0^1 e^{-x^2} d x$$

$$\int_0^1 e^{-x^2} d x \approx e^{-(0.25)^2} \cdot 0.25+e^{-(0.5)^2} \cdot 0.25+e^{-(0.75)^2} \cdot 0.25+e^{-1^2} \cdot 0.25$$

## 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。