📚 Finding areas | 求面积
In A-Level Mathematics, integration is one of the most powerful tools for finding the area under a curve. Whether you are dealing with simple polynomials, trigonometric functions, or parametric equations, the definite integral provides a precise way to calculate the area bounded by curves and lines. This article covers everything you need to know for the Edexcel specification, including areas between curves, areas with respect to the y-axis, and numerical methods like the trapezium rule.
在A-Level数学中，积分是求曲线下方面积的最有力工具之一。无论你处理的是简单的多项式、三角函数还是参数方程，定积分都可以精确计算出曲线与直线所围成的面积。本文涵盖了Edexcel考试大纲中你需要掌握的所有内容，包括曲线间的面积、关于y轴的面积，以及梯形法则等数值方法。
1. Definite Integration Recap | 定积分复习
The definite integral of a function f(x) from x = a to x = b is written as ∫ab f(x) dx. It represents the signed area between the curve y = f(x) and the x-axis, from x = a to x = b. To evaluate it, we find an antiderivative F(x) such that F'(x) = f(x), and then compute F(b) − F(a). This is the Fundamental Theorem of Calculus.
函数 f(x) 从 x = a 到 x = b 的定积分记作 ∫ab f(x) dx。它表示曲线 y = f(x) 与 x 轴之间从 x = a 到 x = b 的有向面积。要计算它，我们需要找到原函数 F(x) 使得 F'(x) = f(x)，然后计算 F(b) − F(a)。这就是微积分基本定理。
For example, to find the area under y = x² from x = 1 to x = 3, we integrate: ∫13 x² dx = [⅓ x³]13 = 9 − ⅓ = 26/3.
例如，要计算 y = x² 从 x = 1 到 x = 3 下方的面积，我们求积分：∫13 x² dx = [⅓ x³]13 = 9 − ⅓ = 26/3。
2. Area Bounded by a Curve and the x-axis | 曲线与x轴所围面积
When the curve y = f(x) lies entirely above the x-axis on the interval [a, b], the area A is given directly by A = ∫ab f(x) dx. However, if the function dips below the x-axis, the integral gives a negative value for that portion; the actual area must be calculated by splitting the integral at the roots and taking absolute values.
当曲线 y = f(x) 在区间 [a, b] 上完全位于 x 轴上方时，面积 A 直接由 A = ∫ab f(x) dx 给出。但是，如果函数部分在 x 轴下方，积分在那部分将给出负值；实际面积必须在根处分割积分并取绝对值来计算。
It is essential to sketch the curve first to determine where f(x) changes sign. The total area = ∫ac f(x) dx − ∫cb f(x) dx if f(x) is negative between c and b.
首先画出曲线草图确定 f(x) 变号的位置至关重要。总面积 = ∫ac f(x) dx − ∫cb f(x) dx，如果 f(x) 在 c 和 b 之间为负。
3. Handling Negative Areas | 处理负面积
If you blindly integrate a function that crosses the x-axis, the negative and positive areas will cancel each other out, giving the net signed area rather than the total enclosed area. For example, for y = x(x − 2) from x = 0 to x = 3, the curve is below the x-axis for 0 < x < 2 and above for 2 < x < 3. The total area is |∫02 f(x) dx| + ∫23 f(x) dx.
如果你直接对穿过 x 轴的函数进行积分，负面积和正面积会互相抵消，得到的是净有向面积而不是总的围成面积。例如，对于 y = x(x − 2) 从 x = 0 到 x = 3，曲线在 0 < x < 2 时位于 x 轴下方，在 2 < x < 3 时位于上方。总面积为 |∫02 f(x) dx| + ∫23 f(x) dx。
Always use a sketch and a sign table to avoid mistakes. The area is always a positive quantity.
务必使用草图和符号表来避免错误。面积始终是正量。
4. Area Enclosed by a Curve and the y-axis | 曲线与y轴所围面积
Sometimes it is more convenient to integrate with respect to y. If the curve is given as x = g(y), the area between the curve and the y-axis from y = c to y = d is A = ∫cd g(y) dy. This method is useful when the function is easier to invert, for example, y = √x becomes x = y².
有时对 y 积分更方便。如果曲线由 x = g(y) 给出，那么曲线与 y 轴之间从 y = c 到 y = d 的面积为 A = ∫cd g(y) dy。当函数容易求反函数时这种方法很有用，例如 y = √x 变为 x = y²。
In the Edexcel exam, you might need to find the area bounded by a curve, the y-axis, and two horizontal lines y = c and y = d. Ensure you write the integral in terms of y, not x, and adjust limits accordingly.
在Edexcel考试中，你可能需要找到曲线、y轴和两条水平线 y = c、y = d 所围成的面积。确保你用的是关于 y 的积分，而不是 x，并相应地调整上下限。
5. Area Between Two Curves | 两条曲线之间的面积
The area bounded by two curves y = f(x) and y = g(x) between x = a and x = b is given by A = ∫ab [f(x) − g(x)] dx, provided that f(x) ≥ g(x) on the interval. If the curves intersect, you must find the x-coordinates of the intersection points and integrate the absolute difference over each sub-interval.
两条曲线 y = f(x) 和 y = g(x) 在 x = a 和 x = b 之间所围成的面积由 A = ∫ab [f(x) − g(x)] dx 给出，条件是 f(x) ≥ g(x) 在该区间上成立。如果曲线相交，你必须求出交点的 x 坐标，然后在每个子区间上对绝对差进行积分。
For example, to find the area between y = x² and y = x, first solve x² = x → x = 0, 1. For 0 ≤ x ≤ 1, x ≥ x², so Area = ∫01 (x − x²) dx = [½x² − ⅓x³]_{0}^{1} = 1/6.
例如，求 y = x² 和 y = x 之间的面积，首先解 x² = x 得 x = 0, 1。在 0 ≤ x ≤ 1 上，x ≥ x²，所以面积 = ∫01 (x − x²) dx = [½x² − ⅓x³]_{0}^{1} = 1/6。
6. Determining the Limits of Integration | 确定积分上下限
A critical step in area problems is finding the intersection points of curves or the roots of equations. Set the functions equal and solve. You may need to factorise, use the quadratic formula, or apply numerical methods like the Newton-Raphson method if an exact solution is not required.
面积问题中关键的一步是找到曲线交点或方程的根。令函数相等并求解。你可能需要因式分解、使用求根公式，或者如果不要求精确解，可应用牛顿-拉夫森法等数值方法。
Always check whether the limits are given in the question or if you need to determine them. In many Edexcel questions, you will be asked to sketch the region and then find its area, implying you must find the intersection points yourself.
始终检查题目中是否给定了上下限，或是需要你自己确定。在许多Edexcel题目中，你会被要求画出区域草图然后求面积，这意味着你必须自己找到交点。
7. Area
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