Integration Applications: Typical Exam Questions | 积分应用：典型题型

📚 Integration Applications: Typical Exam Questions | 积分应用：典型题型

Integration is a core topic in IB Mathematics, and its applications appear frequently in both Analysis and Approaches (AA) and Applications and Interpretation (AI) papers. This article reviews the most common exam-style questions, from areas and volumes to differential equations and accumulated change.

积分是IB数学的核心主题，其应用在分析与方法（AA）以及应用与解释（AI）的考试中都频繁出现。本文回顾最常见的考试题型，涵盖面积、体积、微分方程和累积变化等。

1. Area Between a Curve and the x-axis | 曲线与 x 轴围成的面积

To find the area bounded by y = f(x), the x-axis, and the lines x = a and x = b, integrate the absolute value of f(x) over the interval: A = ∫_a^b |f(x)| dx. If the curve lies entirely above the x-axis, the absolute value can be ignored; otherwise, you must split the interval at the roots of f(x).

求由 y = f(x)、x 轴以及直线 x = a 和 x = b 所围成的面积时，应对 f(x) 的绝对值在区间上积分：A = ∫_a^b |f(x)| dx。若曲线完全位于 x 轴上方，可直接去掉绝对值；否则，必须在 f(x) 的零点处分段积分。

For example, the area enclosed by y = x² – 4 and the x-axis from x = -2 to x = 2 is ∫_-2² (4 – x²) dx = 32/3. Notice the integrand is reversed because the curve is below the axis.

例如，曲线 y = x² – 4 与 x 轴在 x = -2 到 x = 2 之间围成的面积为 ∫_-2² (4 – x²) dx = 32/3。注意这里被积函数写成 4 – x²，因为曲线位于轴的下方。

Always sketch the graph or use your GDC to identify whether the function is positive, negative, or crosses zero. Splitting the interval incorrectly is a very common source of lost marks in Paper 2.

务必画出草图或使用GDC判断函数是正、是负还是穿过零点。区间分段错误是Paper 2中最常见的失分点之一。

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

When two curves f(x) and g(x) enclose a region, the area is found by integrating the difference of the functions: A = ∫_a^b (f(x) – g(x)) dx, where f(x) ≥ g(x) throughout [a,b]. The limits a and b are found by solving f(x) = g(x).

当两条曲线 f(x) 和 g(x) 围成一个区域时，面积由两个函数之差的积分给出：A = ∫_a^b (f(x) – g(x)) dx，其中在整个区间 [a,b] 上 f(x) ≥ g(x)。积分上下限 a 和 b 通过解方程 f(x) = g(x) 得到。

If the curves cross within the interval, compute separate integrals for each region where one function stays on top. Always sketch or use your GDC to verify the order of the functions.

若曲线在区间内相交，则需在每个函数保持上方的区域分别计算积分。务必画图或使用GDC验证上下位置。

A typical IB question might ask for the area enclosed by a quadratic and a straight line. Solve the intersection equation first, then subtract the lower curve from the upper curve and integrate.

IB典型题目可能要求求一条二次曲线与一条直线所围成的面积。先解交点方程，然后用上方曲线减下方曲线再积分。

3. Volume of Revolution: x-axis | 旋转体体积：绕 x 轴

Rotating the region under y = f(x) around the x-axis from a to b gives a solid whose volume is V = π ∫_a^b [f(x)]² dx. This is the disk method, used when the region touches the axis of rotation.

将 y = f(x) 下方区域绕 x 轴从 a 到 b 旋转一周所得物体体积为 V = π ∫_a^b [f(x)]² dx。这是圆盘法，适用于区域与旋转轴接触的情形。

If you rotate the region between two curves f(x) and g(x), use the washer method: V = π ∫_a^b (f(x)² – g(x)²) dx, where f(x) ≥ g(x) ≥ 0 on [a,b].

若绕 x 轴旋转的是两条曲线 f(x) 和 g(x) 之间的区域，应用垫片法：V = π ∫_a^b (f(x)² – g(x)²) dx，其中在 [a,b] 上 f(x) ≥ g(x) ≥ 0。

Remember to square the functions before subtracting; many students incorrectly integrate (f(x) – g(x))². Expand carefully and use your GDC to check the final decimal.

请记住先平方再相减；很多学生错误地对 (f(x) – g(x))² 进行积分。展开时务必仔细，并可用GDC检查最终小数结果。

4. Volume of Revolution: y-axis | 旋转体体积：绕 y 轴

When rotating around the y-axis, express the curve as x = g(y) and integrate with respect to y: V = π ∫_c^d [g(y)]² dy, where y ranges from c to d. This method is convenient when the function can be easily solved for x.

绕 y 轴旋转时，将曲线表示为 x = g(y)，并对 y 积分：V = π ∫_c^d [g(y)]² dy，其中 y 从 c 到 d。当函数容易解出 x 时，这个方法较为方便。

Alternatively, the cylindrical shell method gives V = 2π ∫_a^b x f(x) dx for the region under y = f(x) rotated about the y-axis. This method avoids solving for x but requires careful identification of the radius x and height

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