Integrals, area and accumulation · Extension
| 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 terms by increasing the power by one and dividing by the new power. Include a constant for an indefinite integral. Evaluate a definite integral by subtracting antiderivative values at the limits.
- 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.
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
Integrate polynomial terms by increasing the power by one and dividing by the new power
Compare the model with the worked case and explain one change.
Find the integral of 3x² from 0 to 2.
An antiderivative is x³. Subtract its values: 2³-0³=8.
Test a tempting shortcut
- Use the limits in the specified order. The constant cancels for a definite integral but is needed for an indefinite integral. Total positive area may require splitting at an axis crossing.
- 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.
Displacement=[t³-3t] from 0 to 2=8-6=2.
A definite integral always equals the total positive area, even when the graph crosses the axis.
Use the limits in the specified order. The constant cancels for a definite integral but is needed for an indefinite integral. Total positive area may require splitting at an axis crossing.
Interpret a new situation
- This 9260 Extension lesson uses polynomial integration and area. Logarithmic integrals, substitution, parts and differential equations are outside its scope.
- 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.
Velocity changes sign at t=1. The two positive distances are 2 and 4, giving 6.
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
- 9260 · Extension · 3.3. Match the target tier and specification before assigning extensions.
- 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.