📚 Exercise 7D.3: Definite Integrals and Area | 练习 7D.3：定积分与面积
In IB Mathematics, whether you are following the Analysis & Approaches or Applications & Interpretation course, Exercise 7D.3 is a landmark set of problems that connects the algebra of integration with geometry. It challenges you to compute the area bounded by a curve and the x‑axis, the area trapped between two curves, and the total area when the function dips below the axis. Mastering the ideas behind this exercise gives you a powerful toolbox for tackling real‑world problems and secures a solid foundation for the calculus components of your IB exam.
在 IB 数学中，不论你学习的是分析与方法还是应用与解释课程，练习 7D.3 都是一组标志性的问题，它将积分的代数运算与几何意义联系起来。它要求你计算曲线与 x 轴围成的面积、两条曲线之间夹出的面积，以及函数越过 x 轴后需要累加的总面积。掌握这些习题背后的思想，你就拥有了处理现实世界问题的强大工具箱，也为 IB 考试中微积分模块打下了扎实的基础。
1. The Fundamental Theorem of Calculus | 微积分基本定理
The whole business of calculating area with integrals rests on the Fundamental Theorem of Calculus. It states that if F is an antiderivative of a continuous function f on [a, b], then ∫ab y simple equation tells us that integration can be evaluated by antidifferentiation, bypassing the need for infinite Riemann sums.
用积分计算面积全部依赖于微积分基本定理。该定理指出，如果 F 是连续函数 f 在 [a, b] 上的一个原函数，那么 ∫ab f(x) dx = F(b) − F(a)。这个看似简单的等式告诉我们，可以通过求反导数来求定积分，而无需使用无穷黎曼和。
then substitute the upper and lower limits. For example, with f(x) = 3x², F(x) = x³, so ∫13 1³ = 26. Keep the theorem at the front of your mind; every area calculation depends on it.
在做练习 7D.3 时，第一步永远是正确求出原函数。写下 F(x) = ∫ f(x) dx，然后代入上下限。例如 f(x) = 3x²，原函数 F(x) = x³，因此 ∫13 3x² dx = 3³ − 1³ = 26。请时刻记住这一定理，每次面积计算都离不开它。
2. Area Above the x‑axis | x 轴上方的面积
If f(x) ≥ 0 for all x in [a, b], the definite integral ∫ab f(x) dx directly gives the area between the curve y = f(x), the x‑ and x = b. This is the simplest case in Exercise 7D.3 and the one that builds confidence.
若对所有 x ∈ [a, b] 都有 f(x) ≥ 0，定积分 ∫ab f(x) dx 直接给出曲线 y = f(x)、x 轴以及直线 x = a 和 x = b 之间的面积。这是练习 7D.3 中最简单的情况，能够帮助你建立信心。
A typical question might ask: "Find the area enclosed by y = 4 − x², the x‑axis, x = 0 and x = 2." Here, check that f(x) is non‑negative on [0,2]: 4 − x² ≥ 0 for x ≤ 2, so the area = ∫02 (4 − x²) dx = [4x − x³/3]02 8 − 8/3) = 16/3 units². Always verify the sign of f(x) before trusting the integral alone.
典型的问题可能是：“求 y = 4 − x²、x 轴、x = 0 与 x = 2 所围成的面积。”这里需要检查 f(x) 在 [0,2] 上是否非负：当 x ≤ 2 时 4 − x² ≥ 0，因此面积 = ∫02 (4 − x²) dx = [4x − x³/3]02 = (8 − 8/3) = 16/3 平方单位。在单纯依赖积分之前，务必核查 f(x) 的符号。
3. Area Below the x‑axis | x 轴下方的面积
When f(x) ≤ 0 on [a, b], the definite integral ∫ab (x) dx yields a negative number. The genuine geometric area is the absolute value: A = − ∫ab f(x) dx, or equivalently ∫ab [−f(x)] dx. Exercise 7D.3 often includes a part where you must recognise this sign flips.
当 f(x) ≤ 0 在 [a, b] 上时，定积分 ∫ab f(x) dx 得出的结果是一个负数。真正的几何面积是绝对值：A = − ∫ab f(x) dx，或等价地 ∫ab [−f(x)] dx。练习 7D.3 经常会有一问要求你认识这种符号翻转。
−2x from x = 1 to x = 3. Here f(x) is negative throughout. The integral evaluates to [−x²]13 = −9 + 1 = −8. The area is |−8| self to sketch the region first, then decide whether the integral needs a sign adjustment.
考虑直线 y = −2x 从 x = 1 到 x = 3。这里 f(x) 全程为负。积分计算得到 [−x²]13 = −9 + 1 = −8。面积等于 |−8| = 8 平方单位。许多学生忘记负号而失分；训练自己先画出区域草图，再决定积分是否需要调整符号。
4. Total Area with Sign Changes | 带有符号变化的总面积
The most frequently examined scenario in Exercise 7D.3 is a function that crosses the x‑axis one or more times within the interval. You cannot simply compute ∫ab f(x) dx from start to finish, because the positive and negative parts would cancel each other. Instead, split the integral at each root.
练习 7D.3 中最常见的考试情形是函数在区间内穿越 x 轴一次或多次。你不能机械地从头到尾计算 ∫ab f(x) dx，因为正负部分会互相抵消。应当在每个零点处将积分拆开。
For example, find the total area bounded by y = x³ − 4x and the x‑ −2, 0, 2. On [−2,0] the function is positive, and on [0,2] it is negative. Total area = ∫−20 (x³ − 4x) dx + ∫02 h yields 4 + 4 = 8 units². Always present the answer as a positive number and use the word "total area" when the directions ask for it.
例如，求 y = x³ − 4x 与 x 轴在 x = −2 到 x = 2 之间所围成的总面积。零点为 x = −2, 0, 2。在 [−2,0] 上函数为正，在 [0,2] 上为负。总面积 = ∫−20 (x³ − 4x) dx + ∫02 −(x³ − 4x) dx。分别计算得 4 + 4 = 8 平方单位。结果务必以正数呈现，并在题目要求“总面积”时使用绝对值分段求和。
5. Area Between Two Curves | 两条曲线之间的面积
Beyond the x‑ region sandwiched between two curves. If f(x) ≥ g(x) on [a, b], the area between them is ∫ab [f(x) − g(x)] dx. The "top minus bottom" rule works regardless of whether the curves are above or below the axis, as long as the inequality holds.
除了 x 轴，练习 7D.3 还延伸到两条曲线夹住的区域。若在 [a, b] 上有 f(x) ≥ g(x)，则二者之间的面积为 ∫ab [f(x) − g(x)] dx。无论曲线是在轴的上方还是下方，“上减下”的法则始终成立，只要这一不等式成立即可。
which is above: a quick test at x = 0 shows 1 < 3, so the line is on top. Then area = ∫−12 [(x + 3) − (x² + 1)] dx = ∫−12 (2 + x − x²) dx = [2x + x²/2 − x³/3]−12 ing the two graphs is the best way to avoid the common mistake of swapping top and bottom.
设想 y = x² + 1 与 y = x + 3 在它们相交的区间上，比如从 x = −1 到 x = 2。比较谁在上方：在 x = 0 处检验，1 < 3，因此直线在上。于是面积 = ∫−12 [(x + 3) − (x² + 1)] dx = ∫−12 (2 + x − x²) dx = [2x + x²/2 − x³/3]−12 = 9/2 平方单位。画一张两条曲线的草图，是避免弄混“上减下”的最好方法。
6. Finding Points of Intersection | 求交点
g(x). These solutions become the boundaries of the enclosed region. Accurate algebra here is crucial; a mistake in the intersection points will ruin your area calculation.
当积分上下限没有直接给出时，练习 7D.3 要求你通过解方程 f(x) = g(x) 来自行求出。这些解就是所围区域的边界。此处的代数运算必须精确；交点的错误会导致整个面积计算功亏一篑。
= 0, the parabola gives 6 while the line gives 2, so the parabola is on top. Then area = ∫−22 [(6 + x − x²) − (x + 2)] dx = ∫−22 (4 − x²) dx = [4x − x³/3]−22 = (8 − 8/3) − (−8 + 8/3) = 32/3 units². Always check the "topness" in the middle of the interval.
对于 y = 6 + x − x² 与 y = x + 2 给出的曲线，令 6 + x − x² = x + 2 → 4 − x² = 0 → x = ±2。区域由 x = −2 和 x = 2 围成。在这两个根之间判断谁大：在 x = 0 处，抛物线值为 6，直线为 2，所以抛物线在上。于是面积 = ∫−22 [(6 + x − x²) − (x + 2)] dx = ∫−22 (4 − x²) dx = [4x − x³/3]−22 = (8 − 8/3) − (−8 + 8/3) = 32/3 平方单位。永远要在区间中间点处检验“谁在上方”。
7. Splitting the Integral for Piecewise Functions | 分段函数的积分分割
Sometimes the top‑bottom relationship changes within the region. For instance, two curves may cross inside the interval, so the "top" function swaps. Exercise 7D.3 questions that feature a third intersection point will require you to split the area integral at that crossing point.
有时候，在所求区域内部，上下位置关系会发生变化。比如两条曲线可能在区间内交叉，导致“上方”函数发生交换。练习 7D.3 中如果出现第三个交点，就需要在那个交点处将面积积分分割。
Consider the curves y = x² and y = √x from x = 0 to x = 2. They intersect at x = 0 and x = 1. Between 0 and 1, √x ≥ x² (since at x = 0.25, √0.25 = 0.5, x² = 0.0625). Between 1 and 2, x² ≥ √x. Thus total area = ∫01 (√x − x²) dx + ∫12 (x² − √x) dx. Evaluating gives 1/3 + (8/3 − (2/3)2√2 .333 + 0.781 = 1.114 units². Splitting integrals correctly is a hallmark of strong problem‑solving.
考虑曲线 y = x² 和 y = √x 从 x = 0 到 x = 2。两曲线在 x = 0 和 x = 1 处相交。在 0 与 1 之间，√x ≥ x²（因为 x = 0.25 时，√0.25 = 0.5，x² = 0.0625）。在 1 与 2 之间，x² ≥ √x。因此总面积 = ∫01 (√x − x²) dx + ∫12 (x² − √x) dx。计算得到 1/3 + (8/3 − (2/3)2√2) ≈ 0.333 + 0.781 = 1.114 平方单位。正确分割积分是扎实解题能力的标志。
8. Area with Respect to the y‑axis | 关于 y 轴的面积
ines, swapping variables can simplify the calculation. The area between the curve and the y‑axis from y = c to y = d is ∫cd x dy, where x must be expressed in terms of y.
练习 7D.3 中，有些区域用 y 的函数来描述更加自然。当曲线以 x = g(y) 的形式给出，或区域边界为水平线时，交换变量可以简化计算。曲线与 y 轴之间从 y = c 到 y = d 的面积为 ∫cd x dy，其中 x 需用 y 表示。
1 and the lines y = −1, y = 2. Instead of solving for y, just integrate directly: area = ∫−12 (y² + 1) dy = [y³/3 + y]−12 − 1) = (8/3 + 2) − (−4/3) = 6 units². The same technique works for the area between two x‑functions of y: if u(y) ≥ v(y) on [c, d], area = ∫cd [u(y) − v(y)] dy.
例如，求抛物线 x = y² + 1 与直线 y = −1, y = 2 围成的面积。不必将 y 解出来，直接积分即可：面积 = ∫−12 (y² + 1) dy = [y³/3 + y]−12 = (8/3 + 2) − (−1/3 − 1) = (8/3 + 2) − (−4/3) = 6 平方单位。同样方法适用于用 y 表示的两条曲线之间的面积：若在 [c, d] 上 u(y) ≥ v(y)，则面积 = ∫cd [u(y) − v(y)] dy。
9. Using Technology: GDC Tips | 使用技术：图形计算器技巧
Your graphical display calculator (GDC) is permitted in most IB exams and is invaluable for Exercise 7D.3. Use it to visualise the region, find intersections, and compute definite integrals numerically. However, you must still show the setup analytically to earn method marks.
你的图形计算器在大多数 IB 考试中都是允许使用的，对练习 7D.3 来说尤为宝贵。可以用它观察区域、求交点并数值计算定积分。不过，你仍然需要写出解析的设置过程，以获取方法分数。
es on the GDC to confirm the number of intersections and which function is on top. Use the "intersect" or "analyze graph" tool to find the x‑coordinates. Then write the integral expression on paper, e.g. Area = ∫ab (top − bottom) dx, and finally use the calculator's integration function to obtain the numeric value. Round to three significant figures unless otherwise instructed. Never just write a number without the integral statement; examiners want to see your reasoning.
一个良好的工作流程是：先在计算器上画出曲线草图，确认交点个数以及哪条曲线在上。使用“交点”或“图形分析”工具求出 x 坐标。然后在纸上写出积分表达式，例如 Area = ∫ab (top − bottom) dx，最后用计算器的积分功能得出数值。除非另有要求，结果保留三位有效数字。千万不要只写一个数字而不写积分式；阅卷人希望看到你的推理过程。
10. Common Pitfalls and How to Avoid Them | 常见陷阱与避免方法
Even strong students lose points on Exercise 7D.3 by falling into predictable traps. Watch out for the following: forgetting to check if the function is negative, mixing up the order in "top minus bottom", failing to split the integral when curves cross, and forgetting to add the absolute values of the pieces when finding total area.
即便是能力很强的学生，也会在练习 7D.3 中因一些常见的陷阱而丢分。请注意以下几点：忘记检查函数是否为负、混淆“上减下”的顺序、当曲线交叉时未能拆分积分，以及在求总面积时忘记将各部分的绝对值相加。
Another classic mistake is to write the area as ∫ab rts where f(x) ≥ 0 and where f(x) ≤ 0. Then build the sum of integrals accordingly. Double‑check your antiderivative by differentiating it before plugging in limits.
另一个经典错误是当 f(x) 符号变化时仍将面积写作 ∫ab f(x) dx，导致正负抵消。要避免这一点，务必先在区间内解 f(x) = 0。标出 f(x) ≥ 0 和 f(x) ≤ 0 的区段，然后据此构造积分之和。代入上下限之前，先对原函数求导以检验其正确性。
11. IB Exam‑Style Questions Overview | IB 考试风格问题概览
In the IB Mathematics Analysis & Approaches SL and HL exams, area questions from exercises like 7D.3 appear in both Paper 1 (non‑calculator) and Paper 2 (calculator). A typical 7‑mark question might ask you to find the area enclosed by a cubic function and a line, or to determine a parameter so that a certain area equals a given value.
在 IB 数学分析与方法的 SL 和 HL 考试中，类似练习 7D.3 的面积问题会同时出现在试卷一（非计算器）和试卷二（允许计算器）中。一道典型的 7 分题可能会要求你求出三次函数与直线围成的面积，或确定某个参数使得特定面积等于给定值。
For Applications & Interpretation, the context may be linked to modelling: for instance, revenue and cost curves where the area between them represents profit or surplus. The underlying mathematics is identical, but you need to interpret the result in context. Practice with real past papers where you have to identify the domain, set up the integral, and give the answer with correct units.
对于应用与解释课程，题目背景可能与建模有关：例如收入曲线与成本曲线，它们之间的面积代表利润或剩余。数学内核是相同的，只不过需要结合背景解释结果。建议用真题进行练习，要求你确定定义域、建立积分，并给出带有正确单位的答案。
12. Summary and Key Takeaways | 总结与关键要点
Exercise 7D.3 teaches you that definite integrals are the gateway to measuring area. The golden rules are: sketch the region, find the intercepts, decide which curve is on top, handle sign changes with absolute parts, and always present your final area as a positive quantity. Whether you are working with polynomial, trigonometric, exponential or radical functions, the procedure remains the same.
练习 7D.3 教会你定积分是测量面积的门户。黄金法则是：画出区域的草图，求出截距，确定哪条曲线在上，用绝对值处理符号变化的部分，并且始终以正数呈现最终面积。无论面对的是多项式、三角函数、指数函数还是根式函数，解题步骤始终如一。
As you move through the exercise, focus on building fluency with antiderivatives and on communicating your method clearly. The IB exam rewards structured solutions that explain the splitting of integrals and the reasoning behind "top minus bottom". With systematic practice, you will be able to tackle any 7D.3‑style problem swiftly and accurately, laying the groundwork for the more advanced integration topics ahead.
在做练习的过程中，要专注于熟练求原函数并清晰地呈现解题方法。IB 考试青睐结构分明的解答，其中明确说明了积分的拆分以及“上减下”的理由。通过系统练习，你将能够又快又准地解决任何 7D.3 风格的问题，为后续更高级的积分专题打下坚实基础。
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