📚 Definite Integrals for Finding Area | 定积分求面积方法
The definite integral is one of the most powerful tools in calculus. In addition to measuring accumulated change, it can be used to calculate the area of plane regions bounded by curves. This article explains the standard methods for using definite integrals to find area, with worked examples and examination tips.
定积分是微积分中最有力的工具之一。它不仅可以计算累计变化量，还可以用来求由曲线围成的平面区域面积。本文将系统讲解用定积分求面积的标准方法，并结合实例与考试建议帮助你掌握这一重要考点。
1. The Fundamental Idea | 基本思想
The definite integral ∫ab f(x) dx represents the signed area between the graph y = f(x) and the x-axis over the interval [a, b]. It is defined as the limit of Riemann sums as the partition becomes finer.
定积分 ∫ab f(x) dx 表示曲线 y = f(x) 与 x 轴在区间 [a, b] 上围成的有向面积。它的定义是黎曼和当分割无限加密时的极限。
∫ab f(x) dx = limn→∞ Σi=1n f(xi) Δx
“Signed” means that regions above the x-axis contribute positive area, while regions below the x-axis contribute negative area. This distinction is essential when calculating true geometric area.
“有向”意味着 x 轴上方的区域提供正面积，而 x 轴下方的区域提供负面积。在计算真正的几何面积时，这一区别至关重要。
2. Area Above the x-axis | x 轴上方的面积
If f(x) ≥ 0 for all x in [a, b], then the geometric area is exactly equal to the definite integral.
如果在 [a, b] 上恒有 f(x) ≥ 0，那么几何面积恰好等于定积分。
Area = ∫ab f(x) dx
For example, for f(x) = x² from x = 0 to x = 2, the curve lies entirely above the x-axis. Therefore the area is
例如，函数 f(x) = x² 在 x = 0 到 x = 2 之间完全位于 x 轴上方，因此面积为
∫02 x² dx = [ x³ / 3 ]02 = 8 / 3
The area is 8/3 square units.
该面积为 8/3 平方单位。
3. Area Below the x-axis | x 轴下方的面积
If f(x) ≤ 0 for all x in [a, b], then the definite integral is negative. The actual geometric area is the absolute value of the integral.
如果在 [a, b] 上恒有 f(x) ≤ 0，那么定积分为负数。实际的几何面积应取积分结果的绝对值。
Area = − ∫ab f(x) dx
For instance, consider f(x) = −x² on [0, 1]. The integral is −1/3, so the shaded area between the curve and the x-axis is 1/3 square units.
例如，考虑 f(x) = −x² 在 [0, 1] 上的情况。定积分为 −1/3，因此曲线与 x 轴之间的阴影面积为 1/3 平方单位。
4. Total Area When the Curve Crosses the x-axis | 曲线穿过 x 轴时的总面积
When f(x) changes sign inside [a, b], integration must be split at every x-intercept. On each subinterval, use the absolute value of f(x) before integrating.
当 f(x) 在 [a, b] 内变号时，必须在每一个与 x 轴的交点处分段积分。在每个子区间上，应使用 |f(x)| 再进行积分。
Area = ∫ac₁ |f(x)| dx + ∫c₁c₂ |f(x)| dx + … + ∫cₙb |f(x)| dx
Here c₁, c₂, …, cₙ are the roots of f(x) between a and b.
其中 c₁, c₂, …, cₙ 是 f(x) 在 a 与 b 之间的根。
For example, ∫02π sin x dx = 0, but the total area between y = sin x and the x-axis is not zero. Since sin x is positive on (0, π) and negative on (π, 2π), the true area is 4 square units.
例如，∫02π sin x dx = 0，但 y = sin x 与 x 轴之间的总面积并不是 0。因为 sin x 在 (0, π) 上为正，在 (π, 2π) 上为负，所以真正的面积是 4 平方单位。
5. Area Between Two Curves | 两曲线之间的面积
Suppose two functions f and g satisfy f(x) ≥ g(x) for all x in [a, b]. The area of the region between the two curves is the integral of the upper function minus the lower function.
设两个函数 f 与 g 在 [a, b] 上满足 f(x) ≥ g(x)，则两条曲线之间的区域面积等于上函数减下函数的积分。
Area = ∫ab [ f(x) − g(x) ] dx
For example, y = x² and y = x³ intersect at x = 0 and x = 1. On (0, 1), x² ≥ x³, so
例如，y = x² 与 y = x³ 在 x = 0 和 x = 1 处相交。在 (0, 1) 上，x² ≥ x³，因此
Area = ∫01 (x² − x³) dx = 1/3 − 1/4 = 1/12
The area is 1/12 square units.
该面积为 1/12 平方单位。
6. Finding Intersection Points and Choosing the Upper Curve | 求交点并确定上函数
Before applying the area formula, you must know the order of the two curves. Intersection points are found by solving f(x) = g(x).
在应用面积公式之前，必须判断两条曲线的上下位置关系。交点通过解方程 f(x) = g(x) 得到。
If the order changes inside the interval, split the interval at the intersection points and integrate each piece separately.
如果两条曲线的上下关系在区间内发生变化，则应以交点为界，将区间分开，再分别积分。
For example, y = sin x and y = cos x meet at x = π/4 on [0, π/2]. Before π/4, cos x is above; after π/4, sin x is above. The area must be calculated in two parts.
例如，y = sin x 与 y = cos x 在 [0, π/2] 上相交于 x = π/4。在 π/4 之前 cos x 在上方；在 π/4 之后 sin x 在上方。因此面积必须分两部分计算。
7. Integrating with Respect to y | 对 y 积分的情形
Some regions are easier to describe when x is written as a function of y. If the right boundary is x = R(y) and the left boundary is x = L(y), then for y from c to d,
有些区域用 x 关于 y 的函数来描述更简便。如果右边界为 x = R(y)，左边界为 x = L(y)，则当 y 从 c 到 d 时，
Area = ∫cd [ R(y) − L(y) ] dy
Consider x = y² and x = y + 2. Solving y² = y + 2 gives y = −1 and y = 2. On this interval, the line is on the right and the parabola is on the left.
考虑 x = y² 与 x = y + 2。解 y² = y + 2 得 y = −1 与 y = 2。在这个区间内，直线在右侧，抛物线在左侧。
∫−12 (y + 2 − y²) dy = [ y²/2 + 2y − y³/3 ]−12 = 9/2
The area enclosed by the two curves is 9/2 square units.
这两条曲线所围成的面积为 9/2 平方单位。
8. Worked Example: Parabola and Line | 实例：抛物线与直线
Find the area enclosed by y = x² and y = 2x + 3.
求由 y = x² 与 y = 2x + 3 所围成的面积。
First find the intersections:
首先求交点：
x² = 2x + 3 ⇒ x² − 2x − 3 = 0 ⇒ x = −1 or x = 3
On the interval [−1, 3], the line lies above the parabola. Therefore,
在区间 [−1, 3] 上，直线位于抛物线上方。因此，
Area = ∫−13 (2x + 3 − x²) dx
The antiderivative is x² + 3x − x³/3. Evaluating:
其原函数为 x² + 3x − x³/3。代入计算：
At x = 3: 9 + 9 − 9 = 9
At x = −1: 1 − 3 + 1/3 = −5/3
Area = 9 − ( −5/3 ) = 32/3
The enclosed area is 32/3 square units.
所围成的面积为 32/3 平方单位。
9. Worked Example: Sine and Cosine Curves | 实例：正弦与余弦曲线
Find the area bounded by y = sin x and y = cos x on the interval [0, π/2].
求在区间 [0, π/2] 上由 y = sin x 与 y = cos x 所围成的面积。
The two curves intersect at x = π/4. On [0, π/4], cos x is above sin x. On [π/4, π/2], sin x is above cos x.
两条曲线相交于 x = π/4。在 [0, π/4] 上，cos x 在 sin x 上方；在 [π/4, π/2] 上，sin x 在 cos x 上方。
Area = ∫0π/4 (cos x − sin x) dx + ∫π/4π/2 (sin x − cos x) dx
First integral:
第一个积分：
[ sin x + cos x ]0π/4 = √2 − 1
Second integral:
第二个积分：
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