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.
Browse grades and teaching videos01 · Read and understand
What you will learn
- Apply the fundamental theorem with a variable upper bound.
- Include the upper bound's derivative.
- Check the result by integrating first.
Before you start
Antiderivatives, the chain rule, and differentiating powers.
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
A dimensionless accumulation model is A(x) = ∫₀^(x²)(1 + u) du for x ≥ 0. Find A′(2), and explain the extra factor that appears.
Why this math matters
Accumulation and rate are inverse viewpoints, but a moving endpoint adds another layer. Increasing x by a small amount changes the endpoint x² by an amount controlled by 2x. The accumulation responds both to the height of the integrand at that endpoint and to how quickly the endpoint moves.
Set up the model
A useful answer starts with clear assumptions:
- The integrand 1 + u is continuous on every interval involved.
- x is nonnegative, and all variables are dimensionless in this model.
- u is a dummy integration variable, distinct from the input 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.
What changes when an integral's endpoint also changes?
PausedQuestion: Start with the question. Paused.
Question
Start with the question
A dimensionless accumulation model is A(x) = ∫₀^(x²)(1 + u) du for x ≥ 0. Find A′(2), and explain the extra factor that appears.
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.
Identify the outer accumulation
F(z) = ∫₀ᶻ(1 + u) du ⇒ F′(z) = 1 + z
The fundamental theorem recovers the continuous integrand at the upper endpoint. Here A(x) is F evaluated at z = x².
Differentiate the composition
A′(x) = F′(x²) × 2x = (1 + x²)2x
The endpoint speed contributes the factor 2x. Merely replacing u with x² would omit that necessary factor.
Evaluate and independently check
A′(2) = 5 × 4 = 20; A(x) = x² + x⁴/2
Integrating directly gives x² + x⁴/2, whose derivative is 2x + 2x³. At two, this also equals twenty.
The result
A′(2) = 20 units of accumulated output per unit x.
A small positive change Δx near two gives an approximate accumulation change of 20Δx. This is a local approximation, not an exact formula for large changes. If both bounds depend on x, both endpoints contribute with opposite signs.
Common mistakes to catch
- Do not confuse the dummy variable u with the endpoint x².
- Replacing the integrand's input is not enough: differentiate the moving bound.
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 B′(x) for B(x) = ∫₀^(3x) cos u du, using radians.
Show a hint
Evaluate cosine at the upper bound and multiply by its derivative.
Reveal answer and explanation
3 cos(3x)
The upper bound changes at rate three; the fundamental theorem and chain rule give the product.
Practice 2
Find C′(x) for C(x) = ∫ₓ² u² du.
Show a hint
The lower bound moves while the upper bound stays fixed.
Reveal answer and explanation
−x²
C(x) = 8/3 − x³/3, so differentiating gives −x². A moving lower endpoint subtracts accumulation.
Take the idea with you
Sketch what a moving integration endpoint adds or removes before writing its derivative.
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: How close is a short polynomial to an exponential?
Completion is your own study record, not a test score. It stays in this browser, does not sync to another device, and can be removed by clearing browser data.