Integrals, area and accumulation
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| antiderivative/ˌæntɪdɪˈrɪvətɪv/ | nguyên hàm |
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
- 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.
For v(t)=3t²-3 on 0≤t≤2, why are displacement and total distance different?
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
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
For n≠-1, the integral of ax^n is ax^(n+1)/(n+1)+C
Compare the model with the worked case and explain one change.
Find the integral of 3x² from 0 to 2.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
An antiderivative is x³. Subtract its values: 2³-0³=8.
Where it goes wrong
- 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.
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.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
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.
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
True. Area below the axis counts as negative, so a definite integral can be smaller than the total area, or even zero.
Applying it
- 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.
For that motion, find total distance from t=0 to 2.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Velocity changes sign at t=1. The two positive distances are 2 and 4, giving 6.
Match each integration fact to its statement.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Integration reverses differentiation; the constant belongs to indefinite integrals, and signed area must be split when the curve crosses the axis.
In your course
- AQA A-level Mathematics (7357), specification section Q.
- 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.
- For n≠-1, the integral of ax^n is ax^(n+1)/(n+1)+C.
- Watch for: the integral of 1/x is ln|x|+C, not the power rule with n=-1.