📚 Exercise 25G: Definite Integrals and Area Between Curves | 练习25G：定积分与曲线间面积
In IB Mathematics, particularly in the Analysis and Approaches (AA) course, Exercise 25G typically revisits one of the most powerful applications of definite integrals — calculating the area bound by curves. Whether you are working with a simple polynomial, an exponential function, or two intersecting trigonometric graphs, the ability to set up and evaluate a definite integral is essential for Paper 2 and the internal exploration. This exercise sharpens your understanding of the Fundamental Theorem of Calculus and pushes you to think about area as a net accumulation that may involve sign changes, absolute values, and careful limit choices.
在IB数学中，特别是分析与解释(AA)课程里，练习25G通常重温定积分最重要的应用之一——计算由曲线围成的面积。无论你是在处理简单的多项式、指数函数，还是两条相交的三角函数图像，正确设置并计算定积分的能力对于试卷二和内部评估都至关重要。这个练习能加深你对微积分基本定理的理解，并促使你把面积看作可能涉及符号变化、绝对值和谨慎极限选择的一种净累积量。
1. The Definite Integral as a Net Accumulator | 作为净累积器的定积分
presents the signed area between the graph of f and the x-axis over the interval [a, b]. If f(x) is above the axis, the area counts positively; if below, it is interpreted as negative. This signed nature means that before calculating the actual geometric area, we must always consider where the curve crosses the x-axis.
函数f(x)从a到b的定积分，记作∫ₐᵇ f(x) dx，表示f的图像与x轴在区间[a, b]上围成的带符号面积。如果f(x)在轴上方，面积计为正；如果在下方，则视为负值。这种符号特性意味着在计算实际几何面积之前，我们必须始终考虑曲线与x轴相交的位置。
2. The Fundamental Theorem of Calculus (FTC) | 微积分基本定理
To evaluate ∫ₐᵇ f(x) dx, the FTC states that if F is any antiderivative of f (i.e., F'(x) = f(x)), then the integral equals F(b) – trigonometric functions, often using substitution or integration by parts in HL questions.
为了计算∫ₐᵇ f(x) dx，微积分基本定理指出，如果F是f的任意一个原函数（即F'(x) = f(x)），那么积分等于F(b) – F(a)。你必须熟练掌握多项式、指数函数、对数函数和三角函数的原函数求法，HL题中还会经常用到换元法或分部积分法。
3. Area Between a Curve and the x-Axis | 曲线与x轴之间的面积
When a curve f(x) does not change sign on [a, b], the geometric area is simply |∫ₐᵇ f(x) dx| w the axis, the integral alone gives net area, not total area. A classic IB question asks: “Find the total area enclosed by y = x² – 4x + 3 and the x-axis from x=0 to x=3.” =3, and it is positive on [0,1] but negative on [1,3]. The total area is therefore ∫₀¹ (x²–4x+3) dx + |∫₁³ (x²–4x+3) dx|.
当曲线f(x)在[a, b]上符号不变时，几何面积就是|∫ₐᵇ f(x) dx|。如果曲线落到x轴下方，单独的积分给出的是净面积，而非总面积。一道经典的IB题目会问：“求y = x² – 4x + 3与x轴在x=0到x=3之间围成的总面积。”此时二次函数在x=1和x=3处有根，并且在[0,1]上为正，在[1,3]上为负。因此总面积是∫₀¹ (x²–4x+3) dx + |∫₁³ (x²–4x+3) dx|。
4. Splitting the Integral at Intercepts | 在截点处拆分积分
those solutions in order. Then integrate separately over the subintervals and take the absolute value of each integral before summing. Never blindly evaluate a single integral from start to end when the function changes sign — this mistake is frequently tested in multiple-choice sections.
这里的关键技能是找出函数在给定区间内的x轴截点。解方程f(x)=0，并将这些解按顺序排列。然后在各个子区间上分别积分，再对每个积分取绝对值后求和。当函数符号改变时，千万不要盲目地从起点到终点直接求一个积分——这种错误经常在选择题部分被考查。
5. Area Between Two Curves | 两条曲线之间的面积
he area bound between two curves y = f(x) and y = g(x) from x = a to x = b is given by ∫ₐᵇ |f(x) – g(x)| ct the lower function from the upper one. If the curves intersect within the interval, the “top” and “bottom” roles switch, so we must split the integral at each intersection point.
两条曲线y = f(x)和y = g(x)在x = a到x = b之间围成的面积由∫ₐᵇ |f(x) – g(x)| dx给出，前提是f(x)和g(x)连续。在IB练习中，我们通常先画出草图，确定哪条曲线在上方，然后用上方的曲线减去下方的曲线。如果两条曲线在区间内相交，“上”和“下”的角色会互换，因此我们必须在每一个交点处拆分积分。
6. Finding Intersection Points | 求交点
x) for the x-coordinates of intersection. This might involve factoring, using the quadratic formula, or solving trigonometric equations. In a GDC-permitted IB paper, you can use the calculator’s “intersect” braically when required. Always verify that the intersection lies within the given bounds.
要准确计算两条曲线之间的面积，你必须解方程f(x) = g(x)求出交点的x坐标。这可能涉及因式分解、使用二次公式或求解三角方程。在允许使用图形计算器的IB试卷中，你可以使用计算器的“交点”功能，但你还是需要按照要求写出代数设定过程。务必验证交点是否在给定的界限内。
7. Setting Up the Integral Correctly | 正确设置积分表达式
A common IB command term is “write down, but do not evaluate, an integral expression for the area.” you correctly identify the integrand and limits. For area between curves, the integrand is “upper minus lower,” he area enclosed by y = sin x and y = cos x from x = π/4 to x = 5π/4 can be expressed as ∫π/45π/4 |sin x – cos x| split the interval at the point where the two functions cross, i.e., where sin x = cos x.
IB常见的指令词是“写出，但不要求计算，表示该面积的积分表达式。”这考查你是否能正确识别被积函数和积分限。对于曲线间的面积，被积函数是“上减下”，积分限要么是给定的x值，要么是交点。例如，由y = sin x和y = cos x从x = π/4到5π/4围成的面积可以表示为∫π/45π/4 |sin x – cos x| dx。随后，你需要在两个函数相等的点，即sin x = cos x处拆分区间。
8. Using Absolute Values to Simplify Expression | 使用绝对值简化表达式
ever, to evaluate by hand, you must eliminate the absolute value by determining where f(x) – quick sketch is invaluable. Do not forget to apply the absolute value correctly: area = ∫|...| dx ≥ 0 always.
在IB的一些评分方案中，带绝对值的积分表达式可以作为最终设定，尤其是在使用计算器时。然而，如果要手动求值，你必须通过判断f(x) – g(x)的正负来消除绝对值。符号图或快速草图极为有用。不要忘记正确应用绝对值：面积 = ∫|...| dx 恒大于等于零。
9. Parametric Curves and Area | 参数曲线与面积
a parametric curve and the x-axis over a parameter interval [t₁, t₂] can be found using ∫t₁t₂ y(t) · x'(t) dt. Similarly, the area between two parametric curves involves careful transformation. This extension of Exercise 25G often appears in Section B questions, demanding a strong conceptual understanding of chain rule integration.
对于IB AA高水平课程（HL），你可能会遇到参数方程x = x(t), y = y(t)。由参数曲线与x轴在参数区间[t₁, t₂]上围成的面积可以用∫t₁t₂ y(t) · x'(t) dt求得。类似地，两条参数曲线之间的面积需要仔细的变换。练习25G的这种拓展内容常出现在B部分题目中，要求对链式法则积分有扎实的概念理解。
10. Area with Respect to the y-Axis | 关于y轴的面积
ion is given as x = h(y). The area between a curve and the y-axis from y = c to y = d is ∫cd |h(y)| dy. The same principle applies for area between two curves x = f(y) and x = g(y): set up ∫ |right – left| dy. Recognizing when to switch variables saves time and reduces algebra.
有时候对y积分更方便，特别是当函数给的是x = h(y)的形式时。曲线与y轴之间从y=c到y=d的面积是∫cd |h(y)| dy。同样的原理也适用于两条曲线x = f(y)和x = g(y)之间的面积：设定∫ |右减左| dy。能够识别何时该转换变量可以节省时间并减少代数运算。
11. Common Pitfalls and How to Avoid Them | 常见陷阱与回避方法
Beware of three common mistakes: (1) Using a single integral for a function that crosses the axis without taking absolute values, (2) subtracting the wrong order of functions in area-between-curves problems, and (3) miscalculating intersection points due to algebraic slip-ups. Always double-check by asking: “Is my final area positive? Does my sketch match the arithmetic?” Moreover, if an integral gives a negative result for an area question, you have likely missed a sign change.
当心三个常见错误：(1) 对于穿过x轴的函数，没有取绝对值而直接用单个积分求面积，(2) 在求曲线间面积时减反了函数的顺序，(3) 由于代数疏忽算错了交点。始终通过反问来复查：“我的最终面积是正数吗？我的草图与计算吻合吗？”此外，如果某个面积题目的积分结果为负，你很可能忽略了一次符号变化。
12. Worked Example: Area Between f(x)=x³ – 3x and g(x)=x | 示例练习：f(x)=x³ – 3x与g(x)=x之间的面积
Consider f(x)=x³ – 3x and g(x)=x. To find the total enclosed area, first find intersections: x³ – 3x = x ⇒ x³ – 4x = 0 ⇒ x(x – 2)(x + 2)=0, giving x = –2, 0, 2. Over [–2,0], which function is on top? Test with x = –1: f(–1)=2, g(–1)=–1, so f > g. Over [0,2], test x=1: f(1)=–2, g(1)=1, so g > f. The total area = ∫-20 (f – g) dx + ∫02 (g – f) dx = ∫-20 (x³ – 4x) dx + ∫02 (–x³ + 4x) dx. Evaluating: [x⁴/4 – 2x²] from –2 to 0 = (0) – (4 – 8)=4; second integral [–x⁴/4 + 2x²] from 0 to 2 = (–4+8) – 0 =4. Total area = 8 square units.
考虑f(x)=x³ – 3x和g(x)=x。要求围成的总面积，首先求交点：x³ – 3x = x ⇒ x³ – 4x = 0 ⇒ x(x – 2)(x + 2)=0，解得x = –2, 0, 2。在[–2,0]上，哪条曲线在上？用x = –1检验：f(–1)=2，g(–1)=–1，因此f > g。在[0,2]上，检验x=1：f(1)=–2，g(1)=1，因此g > f。总面积 = ∫-20 (f – g) dx + ∫02 (g – f) dx = ∫-20 (x³ – 4x) dx + ∫02 (–x³ + 4x) dx。求值：[x⁴/4 – 2x²]从–2到0 = (0) – (4 – 8)=4；第二个积分[–x⁴/4 + 2x²]从0到2 = (–4+8) – 0 =4。总面积=8平方单位。
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