Integrals, area and accumulation · 积分、面积与累积量
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| antiderivative/ˌæntɪdɪˈrɪvətɪv/ | 原函数 | yuán hán shù |
How much change has accumulated?
- A velocity graph tells us motion at an instant. How can we recover the displacement accumulated over time?
- This lesson studies antiderivative 原函数: A function whose derivative equals the given integrand.
Choose the mathematical structure
- For n≠-1, the integral of ax^n is ax^(n+1)/(n+1)+C. A definite integral is signed accumulation. Split at sign changes when total area or distance is required.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines antiderivative? · 下列哪项描述正确定义了原函数？
A function whose derivative equals the given integrand. · 其导数等于给定被积函数的函数。
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
For v(t)=3t²-3 over 0≤t≤2, displacement=[t³-3t]_0^2=2. Since v changes sign at t=1, distance=-[t³-3t]_0^1+[t³-3t]_1^2=2+4=6. The constants cancel only for a definite integral.
Integrals, area and accumulation · 积分、面积与累积量
For n≠-1, the integral of ax^n is ax^(n+1)/(n+1)+C · 当 n≠-1 时，ax^n 的积分为 ax^(n+1)/(n+1)+C。
Compare the model with the worked case and explain one change. · 对比模型与已解案例，并说明其中一处变化。
Find the integral of 3x² from 0 to 2. · 求 3x² 从 0 到 2 的定积分。
An antiderivative is x³. Subtract its values: 2³-0³=8. · 原函数为 x³。代入上下限相减：2³-0³=8。
Test a tempting shortcut
- The integral of 1/x is ln|x|+C, not the power rule with n=-1. Signed area can be zero even when the total enclosed area is positive.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
A definite integral always equals the total positive area, even when the graph crosses the axis. This claim is false. Explain which definition or assumption it violates.
For v=3t²-3, find displacement from t=0 to 2. · 对于 v=3t²-3，求从 t=0 到 2 的位移。
Displacement=[t³-3t] from 0 to 2=8-6=2. · 位移=[t³-3t] 从 0 到 2 =8-6=2。
A definite integral always equals the total positive area, even when the graph crosses the axis. · 定积分始终等于正的总面积，即使图像穿过坐标轴。
The integral of 1/x is ln|x|+C, not the power rule with n=-1. Signed area can be zero even when the total enclosed area is positive. · 1/x 的积分为 ln|x|+C，而非使用 n=-1 的幂法则。当总包围面积为正时，有符号面积仍可能为零。
Interpret a new situation
- Identify what the integral means and include the correct units. A rate measured per second integrates to the underlying quantity, not to another rate. Differentiate an antiderivative to check it.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
For that motion, find total distance from t=0 to 2. · 对于该运动，求从 t=0 到 2 的总路程。
Velocity changes sign at t=1. The two positive distances are 2 and 4, giving 6. · 速度在 t=1 处变号。两段正距离分别为 2 和 4，合计为 6。
Match each part of a complete solution to its purpose. · 将完整解答的每个部分与其目的相匹配。
An assumption justifies the model; a check tests the result; interpretation connects it to the question. · 假设用于论证模型的合理性；检查用于验证结果；解释用于将其与问题建立联系。
Use this in your course
- edexcel IAL pure mathematics; official unit P2. Other-unit enrichment is identified in the scope review; it is not an extra cash-in requirement.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
A function whose derivative equals the given integrand. Choose the relationship, show the method, check its assumptions and interpret the result.