Bilingual edition: English is preserved and Chinese follows each unit. Terminology uses the confirmed v20260916 glossary; automated semantic review remains traceable.
中英双语版：英文原文完整保留，中文紧随对应单元；术语采用 v20260916 确认表，自动语义审校结果可追溯。

Area Under a Curve

曲线下的面积

Formulating the area under a curve is the first step toward developing the concept of the integral. The area under the curve formed by plotting function f(x) as a function of x can be approximated by drawing rectangles of finite width and height f equal to the value of the function at the center of the interval.

对曲线下的面积进行建模是发展积分概念的第一步。通过将函数f(x)作为x的函数绘制出来，可以近似地用有限宽度的矩形来表示曲线下的面积，这些矩形的高度等于区间中点处函数的值。

If the width of the rectangles is made smaller, then the number N is larger and the approximation of the area is better.

如果矩形的宽度变小，那么N的值会更大，面积的近似值也会更准确。

Show area integral for simple geometries.

展示简单几何体的面积积分

Show area approximation of an integral.

显示积分的面积近似
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Integral as Limit of Area

积分作为面积的极限

The approximation to the area under a curve can be made better by making the approximating rectangles narrower. The idea of the integral is to increase the number of rectangles N toward infinity by taking the limit as the rectangle width approaches zero.

通过使近似矩形更窄，可以更准确地逼近曲线下的面积。积分的概念是通过让矩形的宽度趋近于零，使矩形数量N趋向于无穷大，从而提高近似精度。

While the concept of geometrical area is a convenient way to visualize an integral, the idea of integration is much more general than that. Any continuous physical variable can be "chopped up" into infinitesmal increments (differential elements) so that the sum of the product of that "width" and some function approaches an infinite sum. The integral is a powerful tool for modeling physical problems which involve continuously varying quantities.

尽管几何面积的概念是一种方便的可视化积分方式，但积分的概念比这要广泛得多。任何连续的物理变量都可以被“分割”成无限小的增量（微分元素），使得该“宽度”与某个函数的乘积之和趋近于无限级数。积分是一种强大的工具，可用于建模涉及连续变化量的物理问题。

Show area integral examples.

展示面积积分示例
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Area Integral Examples

面积积分示例

Area examples with simple geometry can reinforce the idea of the integral as the area under a curve. For a function which is just a constant a, then the area formed by the function is just a rectangle.

The general conclusion here is that the integral of a constant is just that constant times the variable of integration x.

这里的总体结论是，常数的积分就是该常数乘以积分变量x。

For a function f(x) = ax, the area is a triangle

对于函数 f(x) = ax，面积是一个三角形

The progression is to the general form for the integral of a polynomial in x:

下一步是推导x的多项式积分的一般形式：

Show more complex geometry.

展现更复杂的几何结构
具有简单几何形状的面积示例可以强化积分作为曲线下方面积的概念。对于一个仅仅是常数a的函数，由该函数形成的面积就是一个矩形。
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Area Integral Approximations

面积积分近似

The area under any continuous curve can be approximated by drawing a number of rectangles. The integral is the limit for an infinite number of rectangles.

任何连续曲线下的面积可以近似地通过画出若干个矩形来估算。积分是当矩形数量趋于无穷时的极限。

Show the integral which is the limiting case.

展示积分，这是极限情况。
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Area Integral Example

面积积分示例

Integrals are useful for finding the area under curves which can be approximated by geometrical methods.

This is a polynomial type integral which can be found as the sum of the integrals of its parts.

这是一个多项式类型的积分，可以通过将其各部分的积分相加来求得。

Show the geometrical approximation.

展示几何近似

Application to the average of a function

函数平均值的应用
积分可用于计算曲线下的面积，该面积可以借助几何方法进行近似。
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