Integrals, area and accumulation
| English | 中文 | Pinyin |
|---|---|---|
| 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
- Integrate polynomial powers by increasing the index by one and dividing by the new index. Include an arbitrary constant and use a supplied point to determine it.
- 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?
A function whose derivative equals the given integrand.
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.
An antiderivative of 3x² is x³+C because differentiating x³ gives 3x². If F(1)=5, then 1+C=5, so C=4 and F(x)=x³+4.
Integrals, area and accumulation
Integrate polynomial powers by increasing the index by one and dividing by the new index
Compare the model with the worked case and explain one change.
For F prime(x)=3x², F(1)=5, find C in F(x)=x³+C.
F(1)=1+C=5, so C=4.
Test a tempting shortcut
- An indefinite integral needs a constant. Integrating each term changes its power; copying the derivative rule gives the wrong result.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
Every indefinite integral has only one possible antiderivative. This claim is false. Explain which definition or assumption it violates.
For that F, find F(2).
Use F(x)=x³+4: F(2)=8+4=12.
Every indefinite integral has only one possible antiderivative.
An indefinite integral needs a constant. Integrating each term changes its power; copying the derivative rule gives the wrong result.
Interpret a new situation
- P1 uses indefinite polynomial integration and a point to determine the constant. Definite integrals, areas and the trapezium rule are P2 content.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
For that F, find F(0).
F(0)=0+4=4.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- edexcel IAL mathematics; official unit P1. 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.