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# 数学代写|微积分代写Calculus代考|Defining Surface Area

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## 数学代写|微积分代写Calculus代考|Defining Surface Area

If you revolve a region in the plane that is bounded by the graph of a function over an interval, it sweeps out a solid of revolution, as we saw earlier in the chapter. However, if you revolve only the bounding curve itself, it does not sweep out any interior volume but rather a surface that surrounds the solid and forms part of its boundary. Just as we were interested in defining and finding the length of a curve in the last section, we are now interested in defining and finding the area of a surface generated by revolving a curve about an axis.
Before considering general curves, we begin by rotating horizontal and slanted line segments about the $x$-axis. If we rotate the horizontal line segment $A B$ having length $\Delta x$ about the $x$-axis (Figure 6.28a), we generate a cylinder with surface area $2 \pi y \Delta x$. This area is the same as that of a rectangle with side lengths $\Delta x$ and $2 \pi y$ (Figure $6.28 \mathrm{~b}$ ). The length $2 \pi y$ is the circumference of the circle of radius $y$ generated by rotating the point $(x, y)$ on the line $A B$ about the $x$-axis.
Suppose the line segment $A B$ has length $L$ and is slanted rather than horizontal. Now when $A B$ is rotated about the $x$-axis, it generates a frustum of a cone (Figure 6.29a). From classical geometry, the surface area of this frustum is $2 \pi y^* L$, where $y^=\left(y_1+y_2\right) / 2$ is the average height of the slanted segment $A B$ above the $x$-axis. This surface area is the same as that of a rectangle with side lengths $L$ and $2 \pi y^$ (Figure 6.29b).
Let’s build on these geometric principles to define the area of a surface swept out by revolving more general curves about the $x$-axis. Suppose we want to find the area of the surface swept out by revolving the graph of a nonnegative continuous function $y=f(x), a \leq x \leq b$, about the $x$-axis. We partition the closed interval $[a, b]$ in the usual way and use the points in the partition to subdivide the graph into short arcs. Figure 6.30 shows a typical arc $P Q$ and the band it sweeps out as part of the graph of $f$.

## 数学代写|微积分代写Calculus代考|Work and Fluid Forces

In everyday life, work means an activity that requires muscular or mental effort. In science, the term refers specifically to a force acting on an object and the object’s subsequent displacement. This section shows how to calculate work. The applications run from compressing railroad car springs and emptying subterranean tanks to forcing subatomic particles to collide and lifting satellites into orbit.
Work Done by a Constant Force
When an object moves a distance $d$ along a straight line as a result of being acted on by a force of constant magnitude $F$ in the direction of motion, we define the work $W$ done by the force on the object with the formula
$$W=F d \quad \text { (Constant-force formula for work). }$$
From Equation (1) we see that the unit of work in any system is the unit of force multiplied by the unit of distance. In SI units (SI stands for Système International, or International System), the unit of force is a newton, the unit of distance is a meter, and the unit of work is a newton-meter $(\mathrm{N} \cdot \mathrm{m})$. This combination appears so often it has a special name, the joule. Taking gravitational acceleration at sea level to be $9.8 \mathrm{~m} / \mathrm{sec}^2$, to lift one kilogram one meter requires work of 9.8 joules. This is seen by multiplying the force of 9.8 newtons exerted on one kilogram by the one-meter distance moved. In the British system, the unit of work is the foot-pound, a unit sometimes used in applications. It requires one foot-pound of work to lift a one pound weight a distance of one foot.

## 数学代写|微积分代写Calculus代考|Work and Fluid Forces

$$W=F d \quad \text { (Constant-force formula for work). }$$

## 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。