Integrals, area and accumulation · Extension
הוצג| English | משמעות |
|---|---|
| antiderivative/ˌæntɪdɪˈrɪvətɪv/ | פונקציה מקיימת |
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.
The method
- 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.
For v(t)=3t²-3 on 0≤t≤2, why are displacement and total distance different?
רמת קושי 1/5 · 100 מטבעות + בונוס קטן בהשלמה הראשונה היום
The signed integral over the whole interval gives displacement 2. The motion reverses at t=1, so total distance needs the split: 2 backwards then 4 forwards, totalling 6.
A worked case
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.
רמת קושי 2/5 · 140 מטבעות + בונוס קטן בהשלמה הראשונה היום
An antiderivative is x³. Subtract its values: 2³-0³=8.
Where it goes wrong
- 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.
Not true: A definite integral equals the total positive area, even when the graph crosses the axis. Area below the axis counts as negative, so a definite integral can be smaller than the total area, or even zero.
For v=3t²-3, find displacement from t=0 to 2.
רמת קושי 2/5 · 140 מטבעות + בונוס קטן בהשלמה הראשונה היום
Displacement=[t³-3t] from 0 to 2=8-6=2.
Area below the axis counts as negative, so a definite integral can be smaller than the total area, or even zero.
רמת קושי 1/5 · 100 מטבעות + בונוס קטן בהשלמה הראשונה היום
True. Area below the axis counts as negative, so a definite integral can be smaller than the total area, or even zero.
Applying it
- This 9260 Extension lesson uses polynomial integration and area.
- Logarithmic integrals, substitution, parts and differential equations are outside its scope.
For that motion, find total distance from t=0 to 2.
רמת קושי 2/5 · 140 מטבעות + בונוס קטן בהשלמה הראשונה היום
Velocity changes sign at t=1. The two positive distances are 2 and 4, giving 6.
Match each integration fact to its statement.
רמת קושי 2/5 · 140 מטבעות + בונוס קטן בהשלמה הראשונה היום
Integration reverses differentiation; the constant belongs to indefinite integrals, and signed area must be split when the curve crosses the axis.
In your course
- OxfordAQA International GCSE Mathematics (9260), Extension tier, specification section 3.3.
- Every paper allows a calculator, but method marks still need the expression or equation written before its value.
- Antiderivative: A function whose derivative equals the given integrand.
- Integrate polynomial terms by increasing the power by one and dividing by the new power.
- Watch for: use the limits in the specified order.
Write from a distance of 4
- The velocity is v = t - 2 from t = 0 to t = 4.
- Find the displacement and the distance travelled.
A velocity is v = t - 2 from t = 0 to t = 4. What are the displacement and the distance travelled?
The signed areas on either side of t = 2 cancel for displacement and add for distance.
What is the distance travelled?
רמת קושי 1/5 · 100 מטבעות + בונוס קטן בהשלמה הראשונה היום
The two opposite parts each contribute 2, so the distance is 4. Other options — 6 (the lesson distance for a different velocity); 0 (the displacement used as the distance).