# 数学代写|微积分代写Calculus代写|Length of a Curve y = ƒ(x)

#### Doug I. Jones

Lorem ipsum dolor sit amet, cons the all tetur adiscing elit

couryes™为您提供可以保分的包课服务

couryes-lab™ 为您的留学生涯保驾护航 在代写微积分Calculus方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写微积分Calculus代写方面经验极为丰富，各种代写微积分Calculus相关的作业也就用不着说。

## 数学代写|微积分代写Calculus代写|Length of a Curve y = ƒ(x)

Suppose the curve whose length we want to find is the graph of the function $y=f(x)$ from $x=a$ to $x=b$. In order to derive an integral formula for the length of the curve, we assume that $f$ has a continuous derivative at every point of $[a, b]$. Such a function is called smooth, and its graph is a smooth curve because it does not have any breaks, corners, or cusps.

We partition the interval $[a, b]$ into $n$ subintervals with $a=x_0<x_1<x_2<\cdots<$ $x_n=b$. If $y_k=f\left(x_k\right)$, then the corresponding point $P_k\left(x_k, y_k\right)$ lies on the curve. Next we connect successive points $P_{k-1}$ and $P_k$ with straight-line segments that, taken together, form a polygonal path whose length approximates the length of the curve (Figure 6.22). If we set $\Delta x_k=x_k-x_{k-1}$ and $\Delta y_k=y_k-y_{k-1}$, then a representative line segment in the path has length (see Figure 6.23)
$$L_k=\sqrt{\left(\Delta x_k\right)^2+\left(\Delta y_k\right)^2}$$
so the length of the curve is approximated by the sum
$$\sum_{k=1}^n L_k=\sum_{k=1}^n \sqrt{\left(\Delta x_k\right)^2+\left(\Delta y_k\right)^2}$$
We expect the approximation to improve as the partition of $[a, b]$ becomes finer. In order to evaluate this limit, we use the Mean Value Theorem, which tells us that there is a point $c_k$, with $x_{k-1}<c_k<x_k$, such that
$$\Delta y_k=f^{\prime}\left(c_k\right) \Delta x_k .$$
Substituting this for $\Delta y_k$, the sums in Equation (1) take the form
$$\sum_{k=1}^n L_k=\sum_{k=1}^n \sqrt{\left(\Delta x_k\right)^2+\left(f^{\prime}\left(c_k\right) \Delta x_k\right)^2}=\sum_{k=1}^n \sqrt{1+\left[f^{\prime}\left(c_k\right)\right]^2} \Delta x_k .$$
This is a Riemann sum whose limit we can evaluate. Because $\sqrt{1+\left[f^{\prime}(x)\right]^2}$ is continuous on $[a, b]$, the limit of the Riemann sum on the right-hand side of Equation (2) exists and has the value
$$\lim {n \rightarrow \infty} \sum{k=1}^n L_k=\lim {n \rightarrow \infty} \sum{k=1}^n \sqrt{1+\left[f^{\prime}\left(c_k\right)\right]^2} \Delta x_k=\int_a^b \sqrt{1+\left[f^{\prime}(x)\right]^2} d x$$

## 数学代写|微积分代写Calculus代写|Dealing with Discontinuities in dy,dx

Even if the derivative $d y / d x$ does not exist at some point on a curve, it is possible that $d x / d y$ could exist. This can happen, for example, when a curve has a vertical tangent. In this case, we may be able to find the curve’s length by expressing $x$ as a function of $y$ and applying the following analogue of Equation (3):
Formula for the Length of $x=g(y), c \leq y \leq d$ If $g^{\prime}$ is continuous on $[c, d]$, the length of the curve $x=g(y)$ from $A=(g(c), c)$ to $B=(g(d), d)$ is
$$L=\int_c^d \sqrt{1+\left(\frac{d x}{d y}\right)^2} d y=\int_c^d \sqrt{1+\left[g^{\prime}(y)\right]^2} d y .$$

The Differential Formula for Arc Length
If $y=f(x)$ and if $f^{\prime}$ is continuous on $[a, b]$, then by the Fundamental Theorem of Calculus we can define a new function
$$s(x)=\int_a^x \sqrt{1+\left[f^{\prime}(t)\right]^2} d t .$$
From Equation (3) and Figure 6.22, we see that this function $s(x)$ is continuous and measures the length along the curve $y=f(x)$ from the initial point $P_0(a, f(a))$ to the point $Q(x, f(x))$ for each $x \in[a, b]$. The function $s$ is called the arc length function for $y=f(x)$. From the Fundamental Theorem, the function $s$ is differentiable on $(a, b)$ and
$$\frac{d s}{d x}=\sqrt{1+\left[f^{\prime}(x)\right]^2}=\sqrt{1+\left(\frac{d y}{d x}\right)^2} .$$
Then the differential of arc length is
$$d s=\sqrt{1+\left(\frac{d y}{d x}\right)^2} d x$$
A useful way to remember Equation (6) is to write
$$d s=\sqrt{d x^2+d y^2}$$
which can be integrated between appropriate limits to give the total length of a curve. From this point of view, all the arc length formulas are simply different expressions for the equation $L=\int d s$. Figure 6.27a gives the exact interpretation of $d s$ corresponding to Equation (7). Figure 6.27b is not strictly accurate, but it can be thought of as a simplified approximation of Figure 6.27a. That is, $d s \approx \Delta s$.

# 微积分代考

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

Days
Hours
Minutes
Seconds

# 15% OFF

## On All Tickets

Don’t hesitate and buy tickets today – All tickets are at a special price until 15.08.2021. Hope to see you there :)