Learn with Amar
Teaching video
A teaching recording for this chapter has not been published yet. Start with the worked example below and explore the related animations where available.
Grade 12 chapters and video availability01 · Read and understand
What you will learn
- Split at sign changes when integrating an absolute value.
- Justify the conclusion "Total area=∫₋₁⁰(−x)dx+∫₀¹x dx=1" using the stated assumptions.
Before you start
Definite integrals and areas.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
Compare ∫₋₁¹x dx with the total area between y=x and the x-axis.
Why this math matters
Split at sign changes when integrating an absolute value. This worked micro-lesson connects a precise mathematical condition to a conclusion you can check. The transfer task asks you to change the setting and decide which parts of the reasoning still apply.
Set up the model
A useful answer starts with clear assumptions:
- The curve crosses the axis at zero.
- The geometric area is unsigned.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Distinguish net accumulation from total geometric area
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Compare ∫₋₁¹x dx with the total area between y=x and the x-axis.
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Build the model
The negative and positive triangular contributions are −1/2 and +1/2
An ordinary definite integral records signed area.
Work through the mathematics
∫₋₁¹x dx=0
Symmetric signed contributions cancel.
Check the conclusion
Total area=∫₋₁⁰(−x)dx+∫₀¹x dx=1
Absolute area counts both triangles positively.
The result
Total area=∫₋₁⁰(−x)dx+∫₀¹x dx=1
Absolute area counts both triangles positively.
Common mistakes to catch
- Zero net accumulation does not imply no activity occurred.
- Split absolute-value integrals where the sign changes.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
For velocity v(t)=t on [−1,1], what are net displacement and total distance?
Show a hint
Integrate v and |v| separately.
Reveal answer and explanation
Zero displacement and distance one
Reversing direction can cancel displacement without cancelling travel.
Practice 2
For a continuous function f on [a,b] with a<b, when does its integral equal the integral of |f|?
Show a hint
Compare the signed and unsigned contributions.
Reveal answer and explanation
Exactly when f≥0 throughout the interval
A continuous negative value would produce a negative contribution on a nearby interval, making the unsigned integral strictly larger.
Take the idea with you
Report both signed change and total activity when a quantity reverses direction.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Change integral limits along with the variable
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