Areas Between Curves and Lines | 曲线与直线之间的面积

📚 Areas Between Curves and Lines | 曲线与直线之间的面积

Integration is one of the most powerful tools in A-Level Mathematics, and its geometric application to finding areas between curves and lines is a high-frequency exam topic. In Edexcel Pure Mathematics, you are expected to use definite integrals to compute the area enclosed by a curve and the x-axis, a curve and a line, or two different curves. The key skill is setting up the correct integral by identifying the upper and lower functions and the correct limits.

积分是 A-Level 数学中最强大的工具之一，其在求曲线与直线之间面积方面的几何应用是高频考点。在 Edexcel Pure Mathematics 中，你需要使用定积分计算曲线与 x 轴、曲线与直线或两条曲线所围成的面积。关键技能是通过识别上下函数和正确的积分限，建立正确的积分表达式。

1. The Core Idea | 核心思想

The area between two curves or between a curve and a line is found by summing infinitely many thin vertical rectangles. Each rectangle has height equal to the difference between the upper function and the lower function at that x-value, and width dx. The definite integral from the left intersection point a to the right intersection point b adds up all these rectangles to give the exact area.

两条曲线之间或曲线与直线之间的面积，可以通过对无数个细长竖直矩形求和来求得。每个矩形的高等于该 x 值处上函数与下函数之差，宽为 dx。从左交点 a 到右交点 b 的定积分将所有矩形相加，得到精确面积。

Area = ∫_a^b (upper function − lower function) dx

This idea underpins every area calculation between curves and lines. You should always visualise the region first, then decide which function is on top for the entire interval, or split the interval if needed.

这一思想是所有曲线与直线之间面积计算的基础。你应该首先想象该区域的形状，然后判断在整个区间内哪个函数在上方，如果需要则将区间分割。

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2. Definite Integration as Signed Area | 定积分与带符号面积

A definite integral of a single function f(x) from a to b gives the signed area between the curve and the x-axis. If the curve lies above the x-axis, the integral is positive; if the curve lies below the x-axis, the integral is negative. This means that if the curve crosses the x-axis within the interval, the integral alone does not give the actual geometric area.

单个函数 f(x) 从 a 到 b 的定积分给出曲线与 x 轴之间的带符号面积。如果曲线位于 x 轴上方，积分为正；如果曲线位于 x 轴下方，积分为负。这意味着如果曲线在区间内穿过 x 轴，单独使用积分并不能得到实际的几何面积。

To find the true area between a curve and the x-axis, you must split the integral at the points where the curve crosses the x-axis and take the absolute value of each part, or integrate |f(x)|. For areas between two curves, this issue is avoided by always using the upper function minus the lower function, which is non-negative.

要计算曲线与 x 轴之间的真实面积，你必须在曲线与 x 轴相交处拆分积分，并对每一部分取绝对值，或者对 |f(x)| 进行积分。对于两条曲线之间的面积，可以通过始终使用上函数减去下函数来避免这一问题，因为该差值为非负。

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3. Area Between a Curve and the x-axis | 曲线与 x 轴之间的面积

When finding the area between a curve y = f(x) and the x-axis from x = a to x = b, first check whether f(x) changes sign in the interval. If it does not, then the area is simply the absolute value of the definite integral.

当求曲线 y = f(x) 与 x 轴从 x = a 到 x = b 之间的面积时，首先检查 f(x) 在该区间内是否变号。如果不变号，则面积就是定积分的绝对值。

Area = |∫_a^b f(x) dx|

If f(x) crosses the x-axis at x = c, then you need to compute the area in two parts: |∫_a^c f(x) dx| + |∫_c^b f(x) dx|. This ensures that parts below the x-axis are counted as positive area rather than subtracting from the total.

如果 f(x) 在 x = c 处穿过 x 轴，则需要分两部分计算面积：|∫_a^c f(x) dx| + |∫_c^b f(x) dx|。这样可以确保 x 轴下方的部分被计为正面积，而不是从总面积中减去。

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4. Area Between Two Curves: The Key Formula | 两曲线之间的面积：关键公式

For two functions y = f(x) and y = g(x) that intersect at x = a and x = b, where f(x) ≥ g(x) for all x in [a, b], the area enclosed between them is given by the integral of the difference.

对于相交于 x = a 和 x = b 的两个函数 y = f(x) 和 y = g(x)，如果在 [a, b] 上满足 f(x) ≥ g(x)，则它们之间所围成的面积由二者之差的积分给出。

Area = ∫_a^b [f(x) − g(x)] dx

This formula is extremely versatile: f(x) and g(x) can both be curves, or one can be a straight line. The only requirement is that you know which function is on top. If you are unsure, test a point inside the interval or sketch the graphs.

这个公式非常通用：f(x) 和 g(x) 可以都是曲线，也可以一个是直线。唯一的要求是你要知道哪个函数在上方。如果不确定，可以在区间内测试一个点，或者画出图形。

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5. Finding Intersection Points | 求交点

Limits of integration for the area between two curves or a curve and a line are usually the x-coordinates of their intersection points. To find these points, set the two equations equal to each other and solve for x.

两条曲线或曲线与直线之间面积的积分限通常是它们交点的 x 坐标。要求出这些点，需要令两个方程相等并解出 x。

For example, if y = x² and y = 2x + 3, then equate: x² = 2x + 3. Rearranging gives x² − 2x − 3 = 0, so (x − 3)(x + 1) = 0, giving x = −1 and x = 3. These are the limits a and b in the area integral.

例如，如果 y = x² 和 y = 2x + 3，则令等式相等：x² = 2x + 3。移项得到 x² − 2x − 3 = 0，即 (x − 3)(x + 1) = 0，解得 x = −1 和 x = 3。这些就是面积积分中的下限和上限。

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6. Area Between a Curve and a Line | 曲线与直线之间的面积

A special case of the two-curve formula is when one function is a straight line. For example, to find the area between the curve y = x² and the line y = x + 2, first find the intersection points by solving x² = x + 2, which gives x = −1 and x = 2.

两曲线公式的一个特殊情况是其中一个函数为直线。例如，要求曲线 y = x² 与直线 y = x + 2 之间的面积，首先通过解 x² = x + 2 求交点，得到 x = −1 和 x = 2。

In the interval from −1 to 2, the line lies above the parabola. Therefore, the area is ∫₋₁² [(x + 2) − x²] dx. The integrand is x + 2 − x², which integrates to x²/2 + 2x − x³/3. Evaluating from −1 to 2 gives an area of 9/2 square units.

在从 −1 到 2 的区间内，直线位于抛物线上方。因此，面积为 ∫₋₁² [(x + 2) − x²] dx。被积函数为 x + 2 − x²，其积分为 x²/2 + 2x − x³/3。从 −1 到 2 求值，得到面积为 9/2 平方单位。

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7. Splitting the Area: When Curves Cross | 分割面积：曲线相交时

Sometimes the two functions cross each other inside the interval of interest. In that case, the upper and lower functions swap roles. You cannot use a single integral over the whole interval because the integrand would change sign and cancel out some area.

有时两个函数在感兴趣的区间内相交。此时，上下函数会交换角色。你不能在整个区间上使用单一积分，因为被积函数会变号并抵消部分面积。

The correct method is to split the interval at every crossing point and integrate the absolute difference in each subinterval. For example, if f(x) and g(x) intersect at x = c between a and b, then the total area is ∫_a^c [f(x) − g(x)] dx + ∫_c^b [g(x) − f(x)] dx, assuming f is on top on [a, c] and g is on top on [c, b].

正确的方法是在每个交点处分割区间，并在每个子区间内对绝对差进行积分。例如，如果 f(x) 和 g(x) 在 a 与 b 之间的 x = c 处相交，则总面积为 ∫_a^c [f(x) − g(x)] dx + ∫_c^b [g(x) − f(x)] dx，假设在 [a, c] 上 f 在上方，在 [c, b] 上 g 在上方。

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8. Worked Example 1: Parabola and Line | 例题1：抛物线与直线

Find the area enclosed by the curve y = x² − 4x + 3 and the line y = x − 1.

求曲线 y = x² − 4x + 3 与直线 y = x − 1 所围成的面积。

First, find the intersection points by setting x² − 4x + 3 = x − 1. This rearranges to x² − 5x + 4 = 0, so (x − 1)(x − 4) = 0. The limits are x = 1 and x = 4. Next, determine which function is on top. At x = 2, the line gives y = 1 and the curve gives y = −1, so the line is above the curve. Therefore the area is:

首先，令 x² − 4x + 3 = x − 1 求交点。移项得到 x² − 5x + 4 = 0，即 (x − 1)(x − 4) = 0。积分限为 x = 1 和 x = 4。接下来，判断哪个函数在上方。在 x = 2 处，直线给出 y = 1，曲线给出 y = −1，所以直线在曲线上方。因此面积为：

Area = ∫₁⁴ [(x − 1) − (x² − 4x + 3)] dx = ∫₁⁴ (−x² + 5x − 4) dx

Integrating term by term gives [−x³/3 + 5x²/2 − 4x] evaluated from 1 to 4. At x = 4, the value is −64/3 + 40 − 16 = 8/3. At x = 1, the value is −1/3 + 5/2 − 4 = −11/6. The difference is 8/3 − (−11/6) = 16/6 + 11/6 = 27/6 = 9/2. The required area is 9/2 square units.

逐项积分得到 [−x³/3 + 5x²/2 − 4x]，从 1 到 4 求值。在 x = 4 处，值为 −64/3 + 40 − 16 = 8/3。在 x = 1 处，值为 −1/3 + 5/2 − 4 = −11/6。差为 8/3 − (−11/6) = 16/6 + 11/6 = 27/6 = 9/2。因此所求面积为 9/2 平方单位。

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9. Worked Example 2: Two Parabolas | 例题2：两条抛物线

Find the area bounded by the curves y = x² and y = 2x − x².

求曲线 y = x² 与 y = 2x − x² 所围成的面积。

Set x² = 2x − x², which gives 2x² − 2x = 0, so 2x(x − 1) = 0. The intersection points are x = 0 and x = 1. To decide which curve is on top, test x = 1/2: the first curve gives y = 1/4, while the second gives y = 2(1/2) − (1/2)² = 1 − 1/4 = 3/4. Hence y = 2x − x² is the upper curve. The area is therefore:

令 x² = 2x − x²，得到 2x² − 2x = 0，即 2x(x − 1) = 0。交点为 x = 0 和 x = 1。为了判断哪条曲线在上方，测试 x = 1/2：第一条曲线给出 y = 1/4，第二条给出 y = 2(1/2) − (1/2)² = 1 − 1/4 = 3/4。因此 y = 2x − x² 是上曲线。所以面积为：

Area = ∫₀¹ [(2x − x²) − x²] dx = ∫₀¹ (2x − 2x²) dx

This integrates to [x² − 2x³/3] from 0 to 1. At x = 1, the value is 1 − 2/3 = 1/3; at x = 0, the value is 0. So the area is 1/3 square units. This example shows that even when both curves are non-linear, the same subtract-the-lower-function principle applies.

该积分结果为 [x² − 2x³/3]，从 0 到 1 求值。在 x = 1 处，值为 1 − 2/3 = 1/3；在 x = 0 处，值为 0。因此面积为 1/3 平方单位。这个例子表明，即使两条曲线都是非线性的，同样的“减去下方函数”原则仍然适用。

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10. Common Mistakes and Exam Tips | 常见错误与应试技巧

One of the most common errors is using the wrong order of subtraction. Always subtract the lower function from the upper function. If you reverse the order, the integral becomes negative; taking the absolute value can fix the final answer, but it is better to identify the upper function correctly from the start.

最常见的错误之一是减法的顺序错误。务必用上方函数减去下方函数。如果颠倒顺序，积分会变成负值；对最终答案取绝对值可以补救，但最好从一开始就正确识别上方函数。

Another frequent mistake is forgetting to find the intersection points accurately. If the limits are wrong, the entire area calculation fails. Always solve the equation f(x) = g(x) carefully and check your factors. In Edexcel exams, you may also need exact answers, so keep fractions rather than decimals where possible.

另一个常见错误是未能准确求出交点。如果积分限错误，整个面积计算都会失败。务必仔细求解方程 f(x) = g(x) 并检查因式分解。在 Edexcel 考试中，可能还要求精确答案，因此尽可能保留分数而不是小数。

Finally, when the curves cross within the interval, do not use a single integral without splitting. A quick sign check can save you from losing marks. Sketching a simple graph is always a helpful way to verify the upper and lower functions.

最后，当曲线在区间内相交时，不要在没有拆分的情况下使用单一积分。快速检查符号可以避免丢分。画一个简单的图形始终是验证上下函数的有用方法。

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