Calculus is fundamentally different from the mathematics that you have studied previously: calculus is less static and more dynamic. It is concerned with change and motion; it deals with quantities that approach other quantities. For that reason it may be useful to have an overview of the subject before beginning its intensive study. Here we give a glimpse of some of the main ideas of calculus by showing how the concept of a limit arises when we attempt to solve a variety of problems.
微积分与你以前学过的数学有着本质的不同：微积分是动态的,而非静态，它关注变化与运动，处理变量之间的关系。出于这一考量，此处对该主题的一番概述应有助于日后的深入研究。在这里，通过展示在我们试图解决各种问题的过程中，极限的概念是如何产生的，来一瞥微积分的一些主要思想。
The area problem<面积问题>
The origins of calculus go back at least 2500 years to the ancient Greeks, who found areas using the “method of exhaustion.” They knew how to find the area A of any polygon by dividing it into triangles as in Figure 1 and adding the areas of these triangles.
微积分的起源至少可以追溯到2500年前的古希腊人，他们发明了一种名为“穷竭法”的方法来求得面积。他们通过将多边形划分为若干三角形(如图1所示)并将这些三角形的面积相加来求得多边形的面积A。
Figure 1（图1）
It is a much more difficult problem to find the area of a curved figure. The Greek method of exhaustion was to inscribe polygons in the figure and circumscribe polygons about the figure and then let the number of sides of the polygons increase. Figure 2 illustrates this process for the special case of a circle with inscribed regular polygons.
然而，要求出一个曲边图形的面积就不是那么简单了。希腊的穷竭法是在图形中作一多边形，使之内接于曲边图形，然后让多边形的边数增加。图2展示了内接于圆的正多边形的特殊情况。
Figure 2（图2）
Let An be the area of the inscribed polygon with n sides. As n increases, it appears that An becomes closer and closer to the area of the circle. We say that the area of the circle is the limit of the areas of the inscribed polygons, and we write
设An为内接n边形的面积。随着n的增加，An会变得越来越接近圆的面积。于是，我们说圆的面积是内接多边形的面积的极限，记作
The Greeks themselves did not use limits explicitly. However, by indirect reasoning, Eudoxus (fifth century bc) used exhaustion to prove the familiar formula for the area of a circle:
希腊人并没有明确地使用极限这一概念。然而，通过间接推理，欧多克索斯(公元前5世纪)用穷竭证明了圆的面积公式，即我们所熟知的：
We will use a similar idea in Chapter 4 to find areas of regions of the type shown in Figure 3.We will approximate the desired area A by areas of rectangles (as in Figure), let the width of the rectangles decrease, and then calculate A as the limit of these sums of areas of rectangles.
我们将在第4章中使用类似的方法来求图3所示图形的面积。我们将用矩形的面积来近似所需求的面积A(如图所示)，让矩形的宽减小，然后计算这些矩形面积之和的极限，即为所需求得的面积A。
Figure 3（图3）
The area problem is the central problem in the branch of calculus called integral calculus. The techniques that we will develop in Chapter 4 for finding areas will also enable us to compute the volume of a solid, the length of a curve, the force of water against a dam, the mass and center of gravity of a rod, and the work done in pumping water out of a tank.
面积问题是微积分积分分支中的中心问题。我们将在第四章学习，同时借助这一工具，我们将能够计算固体的体积、曲线的长度、水对大坝的力、杆的质量和重心，以及从水箱中抽水所做的功。
尝试翻译了一点James Stewart的calculus，但是专栏好像不支持插入数学公式。
