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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.
Undergraduate chapters and video availability01 · Read and understand
What you will learn
- Apply the fundamental theorem and chain rule to both integral endpoints.
- Justify the conclusion "F′(x)=2x⁵−x²" using the stated assumptions.
Before you start
Definite integrals and chain rule.
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
Differentiate F(x)=∫ₓ^(x²)t² dt.
Why this math matters
Apply the fundamental theorem and chain rule to both integral endpoints. 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 integrand is continuous for all relevant endpoints.
- Oriented integrals are allowed when x²<x.
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.
Differentiate accumulation with two moving boundaries
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Differentiate F(x)=∫ₓ^(x²)t² dt.
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
F(x)=H(x²)−H(x), where H′(t)=t²
Separate the moving upper and lower limits using an antiderivative.
Work through the mathematics
F′(x)=(x²)²·2x−x²·1
Each endpoint contributes its own chain-rule factor.
Check the conclusion
F′(x)=2x⁵−x²
Direct integration gives F=x⁶/3−x³/3, confirming the derivative.
The result
F′(x)=2x⁵−x²
Direct integration gives F=x⁶/3−x³/3, confirming the derivative.
Common mistakes to catch
- Do not omit the derivative of the upper boundary x².
- The lower endpoint also moves.
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
Find F′(1).
Show a hint
Substitute into the derivative.
Reveal answer and explanation
One
2−1=1.
Practice 2
Why is the lower-endpoint term negative?
Show a hint
Recall upper antiderivative minus lower antiderivative.
Reveal answer and explanation
Increasing the lower boundary removes accumulated area
The sign follows the orientation of the definite integral.
Take the idea with you
Model a quantity collected over a time window whose start and end both change.
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: Track a field along a moving trajectory
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