Calculating the Area Below the x-axis | x轴下方区域面积的计算方法

📚 Calculating the Area Below the x-axis | x轴下方区域面积的计算方法

When studying definite integrals in calculus, one of the most common questions is how to find the actual geometric area enclosed by a curve and the x-axis, especially when the curve lies entirely below the x-axis. The integral gives a signed quantity, so direct evaluation may produce a negative number. To obtain the true area, we must take the absolute value of the definite integral, or adjust the limits of integration appropriately.

在微积分学习定积分时，最常见的问题之一是如何求曲线与x轴围成的实际几何面积，尤其是当曲线完全位于x轴下方时。积分给出的是有符号的量，直接计算可能会得到负数。为了得到真实的面积，我们必须取定积分的绝对值，或适当调整积分区间。

1. Understanding the Area below the x-axis | 理解x轴下方的区域

The region bounded by a curve y = f(x), the x-axis, and the vertical lines x = a and x = b is a classic area problem. If f(x) is negative for every x in [a, b], the curve lies below the x-axis. The geometric area in this case is still a positive quantity, but the definite integral of f(x) from a to b is negative.

由曲线 y = f(x)、x轴以及两条竖直线 x = a 和 x = b 围成的区域是一个经典的面积问题。如果 f(x) 在 [a, b] 上处处为负，那么曲线位于x轴下方。这时的几何面积仍然是一个正数，但 f(x) 从 a 到 b 的定积分却是负数。

Geometrically, “area below the x-axis” means the region extending downward from the x-axis. Visually, we can think of it as the space between the curve and the horizontal axis, but measured downward. When calculating area, no sign is attached; area is always a non-negative real number.

从几何上看，“x轴下方的区域”指的是从x轴向下延伸的部分。直观上，我们可以把它想象成曲线与水平轴之间的空间，但方向向下。计算面积时，不附加符号；面积永远是非负实数。

2. The Signed Area of a Definite Integral | 定积分的符号面积

The definite integral ∫_a^b f(x) dx is defined as the signed area between the graph of f and the x-axis. This means that portions of the graph above the x-axis contribute positive area, while portions below the x-axis contribute negative area.

定积分 ∫_a^b f(x) dx 被定义为函数 f 的图象与x轴之间的“有符号面积”。也就是说，图象在x轴上方的部分贡献正面积，而在x轴下方的部分贡献负面积。

For example, if f(x) = -x² + 1 on [-1, 1], the integral equals the area of the region above the axis. But if we reverse the curve to g(x) = x² – 1 on the same interval, the integral equals a negative number, even though the region has the same geometric size.

例如，若 f(x) = -x² + 1 在 [-1, 1] 上，积分等于x轴上方区域的面积。但如果把曲线反过来，令 g(x) = x² – 1 在同一个区间上，积分却是一个负数，尽管区域的几何大小完全相同。

Signed area = (area above x-axis) − (area below x-axis)

符号面积 = (x轴上方面积) − (x轴下方面积)

3. Why Direct Integration Gives a Negative Value | 为什么直接积分得到负值

Assume f(x) < 0 for all x in [a, b]. By the definition of the definite integral, the Riemann sum is formed by taking values f(xᵢ) times the width Δx. Since every f(xᵢ) is negative, every term in the sum is negative, so the limit is negative. Thus the integral itself is negative.

假设 f(x) < 0 对所有 x ∈ [a, b] 成立。根据定积分的定义，黎曼和由 f(xᵢ) 乘以宽度 Δx 构成。由于每个 f(xᵢ) 都是负数，求和中的每一项都是负的，因此极限也是负的。所以积分本身为负。

Example: ∫₀¹ (−x²) dx = −(1/3). The geometric area enclosed by y = −x², the x-axis, and x = 0 and x = 1 is 1/3, not −1/3.

例如：∫₀¹ (−x²) dx = −(1/3)。由 y = −x²、x轴以及 x = 0 和 x = 1 围成的几何面积是 1/3，而不是 −1/3。

Therefore, to convert a negative integral into a positive area, we must apply the absolute value. For a curve entirely below the x-axis, the area is |∫_a^b f(x) dx| = −∫_a^b f(x) dx.

因此，要把负积分转化为正面积，我们必须取绝对值。对于完全位于x轴下方的曲线，面积等于 |∫_a^b f(x) dx| = −∫_a^b f(x) dx。

4. General Method to Calculate the Area below the x-axis | 计算x轴下方面积的一般方法

If the entire region of interest lies below the x-axis, the general method is simple: first compute the definite integral of f(x) over the given interval, then take its absolute value. Alternatively, since f(x) is negative, integrate −f(x) instead.

如果所考虑的区域完全位于x轴下方，一般方法很简单：先计算 f(x) 在给定区间上的定积分，然后取其绝对值。或者，由于 f(x) 为负，也可以直接对 −f(x) 积分。

More precisely, the geometric area A is given by:

A = ∫_a^b |f(x)| dx

Because |f(x)| = −f(x) when f(x) < 0, this integral is always non-negative. This formula works for both positive and negative functions, making it the universal definition of total area between a curve and the x-axis.

因为当 f(x) < 0 时 |f(x)| = −f(x)，所以这个积分总是非负的。该公式对正函数和负函数都适用，因此是曲线与x轴之间总面积的通用定义。

5. Step-by-Step Procedure | 分步计算步骤

To find the area below the x-axis between x = a and x = b, follow these steps:

要计算x轴下方从 x = a 到 x = b 的面积，请按以下步骤操作：

Check that f(x) ≤ 0 on the entire interval [a, b]. If it is not, split the interval at the zeros of f(x).

确认 f(x) 在整个区间 [a, b] 上满足 f(x) ≤ 0。如果不满足，则需在 f(x) 的零点处分割区间。
Compute the definite integral I = ∫_a^b f(x) dx.

计算定积分 I = ∫_a^b f(x) dx。
Take the absolute value: A = |I| = −I (since I is negative).

取绝对值：A = |I| = −I（因为 I 为负）。
If using |f(x)| directly, integrate ∫_a^b −f(x) dx instead.

如果直接使用 |f(x)|，则改为计算 ∫_a^b −f(x) dx。

Always remember to convert your final answer to a positive number if the original integral was negative.

如果原始积分为负，始终记住将最终答案转换为正数。

6. Dealing with Regions Above and Below the Axis | 处理横跨x轴上下方的区域

A curve often lies partly above and partly below the x-axis within the same interval. In that case, integrating f(x) directly would cancel positive and negative areas. To find total enclosed area, we must split the interval at every root of f(x) in [a, b].

在同一个区间内，曲线常常一部分在x轴上方、一部分在x轴下方。此时直接对 f(x) 积分会抵消正负面积。为了求总围成面积，我们必须按照 f(x) 在 [a, b] 内的每一个根将区间分段。

For example, consider f(x) = x² − 1 on [0, 2]. The roots are x = 1. On [0, 1], f(x) ≤ 0; on [1, 2], f(x) ≥ 0. The total area is:

例如，考虑 f(x) = x² − 1 在 [0, 2] 上。根为 x = 1。在 [0, 1] 上 f(x) ≤ 0；在 [1, 2] 上 f(x) ≥ 0。总面积为：

A = |∫₀¹ (x² − 1) dx| + ∫₁² (x² − 1) dx

Compute each part separately and add the positive values.

分别计算每一段，然后将正值相加。

7. Using Symmetry to Simplify Calculations | 利用对称性简化计算

When the region below the x-axis is symmetric about the y-axis, the integral on [−a, a] can be written as 2∫₀^a f(x) dx. If f(x) is even and negative, the area is simply 2 times the absolute integral on half the interval.

当x轴下方的区域关于y轴对称时，[−a, a] 上的积分可以写成 2∫₀^a f(x) dx。如果 f(x) 是偶函数且为负，那么面积就等于半区间上积分绝对值的 2 倍。

Example: f(x) = −x² + 4 on [−2, 2] is entirely below the axis? Actually f(0) = 4, so it is above axis near the centre. For a below-axis example, use f(x) = x² − 4: it is negative on (−2, 2). Then:

例：f(x) = −x² + 4 在 [−2, 2] 上并非全部位于轴下方？实际上 f(0)=4，所以在中心附近在轴上方。对于轴下方的例子，用 f(x) = x² − 4：它在 (−2, 2) 上为负。那么：

A = 2∫₀² (4 − x²) dx = 2 × [4x − x³/3]₀² = 2 × (16/3) = 32/3

Symmetry reduces computation time and avoids sign errors.

利用对称性可以减少计算时间并避免符号错误。

8. Worked Example 1: Simple Polynomial | 示例1：简单多项式

Find the area enclosed by y = x² − 3x, the x-axis, and the lines x = 1 and x = 2.

求由 y = x² − 3x、x轴以及直线 x = 1 和 x = 2 围成的面积。

First, factor f(x) = x(x − 3). On [1, 2], x > 0 and x − 3 < 0, so f(x) < 0. The area is:

首先，因式分解 f(x) = x(x − 3)。在 [1, 2] 上，x > 0 且 x − 3 < 0，所以 f(x) < 0。面积为：

A = |∫₁² (x² − 3x) dx| = −[(x³/3) − (3x²/2)]₁²

Evaluate: at x = 2, (8/3 − 6) = −10/3; at x = 1, (1/3 − 3/2) = −7/6. The integral is (−10/3) − (−7/6) = −20/6 + 7/6 = −13/6. Thus A = 13/6 square units.

计算：在 x = 2 处，(8/3 − 6) = −10/3；在 x = 1 处，(1/3 − 3/2) = −7/6。积分值为 (−10/3) − (−7/6) = −20/6 + 7/6 = −13/6。因此 A = 13/6 平方单位。

9. Worked Example 2: Trigonometric Function | 示例2：三角函数

Find the area under one arch of y = sin(x) below the x-axis? Actually sin(x) is positive on (0, π). Consider y = −sin(x) on [0, π]. Since −sin(x) ≤ 0, the area is:

求 y = sin(x) 在x轴下方的一个弓形面积？实际上 sin(x) 在 (0, π) 为正。考虑 y = −sin(x) 在 [0, π] 上。由于 −sin(x) ≤ 0，面积为：

A = ∫₀^π sin(x) dx = [−cos(x)]₀^π = −cos(π) + cos(0) = 2

Notice that we integrated −f(x), which gives a positive result directly. This is a common trick when the curve is entirely below the axis: integrate the absolute value.

注意，我们对 −f(x) 积分，直接得到正结果。这是曲线完全位于轴下方时常用的技巧：对绝对值积分。

Another example: find the area bounded by y = cos(x), the x-axis, between x = π/2 and x = π. On this interval cos(x) ≤ 0. So:

另一个例子：求由 y = cos(x)、x轴以及 x = π/2 到 x = π 围成的面积。在此区间上 cos(x) ≤ 0。所以：

A = |∫_π/2^π cos(x) dx| = |[sin(x)]_π/2^π| = |0 − 1| = 1

10. Common Mistakes and How to Avoid Them | 常见错误及避免方法

Forgetting to take the absolute value when the integral is negative. Always check whether the function is non-positive on the interval.

当积分为负时忘记取绝对值。始终检查函数在区间上是否非正。
Integrating f(x) directly over an interval where the function crosses the x-axis. Split the interval at the roots and sum absolute contributions.

在函数穿越x轴的区间上直接对 f(x) 积分。需在根处分割区间，并将各段贡献取绝对值后相加。
Confusing “area” with “definite integral” in applied problems. Area is always positive; integral can be positive, negative, or zero.

在应用问题中混淆“面积”与“定积分”。面积永远是正数；积分可以是正、负或零。
Using incorrect signs when evaluating antiderivatives. Write out the interval separately and simplify carefully.

求原函数时符号错误。应单独写出区间并仔细化简。

11. Summary of the Method | 方法总结

To calculate the area below the x-axis, the core idea is to make the integral positive. The safest universal formula is A = ∫_a^b |f(x)| dx, which automatically handles regions above and below the axis. If you know the entire region is below, compute ∫_a^b f(x) dx and change its sign.

计算x轴下方区域面积的核心思想是让积分变为正数。最保险的通用公式是 A = ∫_a^b |f(x)| dx，它能自动处理x轴上方和下方的区域。如果你知道整个区域都在下方，直接计算 ∫_a^b f(x) dx 并改变其符号。

Remember: split at roots, integrate each piece separately, take absolute values, and sum. With these tools, you can handle any continuous function and any bounded region involving the x-axis.

记住：在根处分割区间，分别对每段积分，取绝对值，再相加。有了这些方法，你就能处理任何连续函数以及任何涉及x轴的有界区域。

Published by TutorHao | 数学 Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Calculating the Area Below the x-axis | x轴下方区域面积的计算方法