Learn with Amar
Teaching video
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Undergraduate chapters and video availability01 · Read and understand
What you will learn
- Integrate speed rather than endpoint displacement.
- Justify the conclusion "L=∫₀^(π/2)3 dt=3π/2" using the stated assumptions.
Before you start
Derivatives and definite integrals.
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
Find the length of r(t)=(3cos t,3sin t) for 0≤t≤π/2.
Why this math matters
Integrate speed rather than endpoint displacement. 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 is traversed once over the stated interval.
- Angles are measured in radians.
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.
Calculate length along a parameterized curve
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Find the length of r(t)=(3cos t,3sin t) for 0≤t≤π/2.
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
r′(t)=(−3sin t,3cos t)
Differentiation gives the instantaneous velocity of the parameterization.
Work through the mathematics
|r′(t)|=3
The identity sin²t+cos²t=1 makes the speed constant.
Check the conclusion
L=∫₀^(π/2)3 dt=3π/2
This quarter-circle arc is longer than its straight endpoint chord.
The result
L=∫₀^(π/2)3 dt=3π/2
This quarter-circle arc is longer than its straight endpoint chord.
Common mistakes to catch
- Arc length integrates speed, not signed velocity.
- An endpoint distance does not measure travel along a curved route.
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 the chord length.
Show a hint
Subtract endpoints (3,0) and (0,3).
Reveal answer and explanation
3√2
Pythagoras gives √(9+9).
Practice 2
Does a faster regular reparameterization change the geometric length?
Show a hint
Substitute in the speed integral.
Reveal answer and explanation
No
The speed change and parameter differential compensate.
Take the idea with you
Compare a curved cable's length with the direct span between its endpoints.
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: Measure how quickly a curve changes direction
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