Using Integration | 使用积分

📚 Using Integration | 使用积分

Integration is one of the two central operations in A-Level Mathematics, and its applications go far beyond reversing differentiation. In the Edexcel Pure Mathematics specification, using integration means calculating areas under curves, areas between curves, volumes of revolution, working with parametric curves, and applying numerical methods such as the trapezium rule. This article explains each application step by step, with the notation and exam techniques you need.

积分是 A-Level 数学的两大核心运算之一，其应用远不止于微分的逆运算。在 Edexcel 纯数学考纲中，使用积分意味着计算曲线下的面积、两条曲线之间的面积、旋转体体积，处理参数曲线，以及应用梯形法则等数值方法。本文将逐步讲解每一种应用，并给出你需要的符号表示和应试技巧。

1. Definite Integrals and Area | 定积分与面积

A definite integral represents the signed area between a curve and the x-axis from x = a to x = b. If the curve lies above the x-axis, the integral is positive; if it lies below, the integral is negative. This signed nature is important in area problems, where you usually want the actual geometric area rather than a signed value.

定积分表示曲线与 x 轴在 x = a 到 x = b 之间的带符号面积。如果曲线位于 x 轴上方，积分为正；如果位于下方，积分为负。这种带符号的性质在面积问题中非常重要，因为通常需要的是实际几何面积，而不是带符号值。

The Fundamental Theorem of Calculus connects integration and differentiation. If F(x) is any antiderivative of f(x), then the definite integral equals F(b) − F(a). This lets you evaluate integrals without constructing limits of Riemann sums.

微积分基本定理将积分与微分联系起来。若 F(x) 是 f(x) 的任意一个原函数，则定积分等于 F(b) − F(a)。这使你可以直接计算定积分，而无需构造黎曼和的极限。

A = ∫ f(x) dx from x = a to x = b = F(b) − F(a)

2. Area Under a Curve Above the x-axis | x 轴上方曲线下的面积

When y = f(x) is non-negative on [a, b], the area between the curve, the x-axis, and the vertical lines x = a and x = b is simply A = ∫ f(x) dx from x = a to x = b. Always write down the integral first and then evaluate it with the correct limits.

当 y = f(x) 在区间 [a, b] 上非负时，曲线、x 轴以及直线 x = a 和 x = b 之间的面积就是 A = ∫ f(x) dx（从 a 到 b）。一定要先写出积分式，再用正确的上下限求值。

For example, to find the area under y = x² + 1 from x = 0 to x = 2, integrate to get ∫ (x² + 1) dx = x³/3 + x. Substituting the limits gives (8/3 + 2) − (0) = 14/3 square units. Include units if the question provides them.

例如，求 y = x² + 1 在 x = 0 到 x = 2 之间的面积：积分得 ∫ (x² + 1) dx = x³/3 + x。代入上下限得 (8/3 + 2) − (0) = 14/3 平方单位。如果题目给出了单位，答案必须包含单位。

A = ∫ (x² + 1) dx from 0 to 2 = [x³/3 + x] from 0 to 2 = 14/3

3. Area Below the x-axis and Splitting the Integral | x 轴下方的面积与拆分积分

If the curve is below the x-axis on part or all of the interval, the integral gives a negative value. To find the actual area, take the absolute value of the integral over each interval where the sign of f(x) is constant, or split the integral at the roots of f(x).

如果曲线在区间的部分或全部位于 x 轴下方，积分会得到负值。为了求实际面积，应在 f(x) 符号保持不变的每个区间上取积分的绝对值，或在 f(x) 的零点处拆分积分。

For example, with y = x² − 4 from x = 0 to x = 3, the curve crosses the x-axis at x = 2. The integral from 0 to 2 is −16/3, and the integral from 2 to 3 is 7/3. The actual area is |−16/3| + 7/3 = 23/3 square units.

例如，对于 y = x² − 4 在 x = 0 到 x = 3 的区间，曲线在 x = 2 处穿过 x 轴。从 0 到 2 的积分为 −16/3，从 2 到 3 的积分为 7/3。实际面积为 |−16/3| + 7/3 = 23/3 平方单位。

Area = ∫ |f(x)| dx = −∫ f(x) dx + ∫ f(x) dx over split intervals

4. Area Between Two Curves | 两条曲线之间的面积

To find the area enclosed by two curves y = f(x) and y = g(x), use A = ∫ [upper function − lower function] dx between their intersection points. First solve f(x) = g(x) to find the limits, and check which function is on top in the interval.

要求两条曲线 y = f(x) 和 y = g(x) 所围成的面积，使用 A = ∫ [上方函数 − 下方函数] dx，积分限为两曲线交点的 x 坐标。首先解方程 f(x) = g(x) 求出交点，并检查在该区间内哪条曲线位于上方。

For example, the curves y = x² and y = 2x intersect at x = 0 and x = 2. On the interval (0, 2), y = 2x is the upper curve. The enclosed area is ∫ (2x − x²) dx from 0 to 2 = [x² − x³/3] from 0 to 2 = 4/3.

例如，曲线 y = x² 与 y = 2x 相交于 x = 0 和 x = 2。在区间 (0, 2)

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