Learn with Amar
Teaching video
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Undergraduate chapters and video availability01 · Read and understand
What you will learn
- Use a full-period integral to distinguish different harmonic modes.
- Justify the conclusion "The integral is zero" using the stated assumptions.
Before you start
Integration and trigonometric product identities.
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
Evaluate ∫₀^(2π) sin x sin 2x dx.
Why this math matters
Use a full-period integral to distinguish different harmonic modes. 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 interval is exactly one full period of the integer modes.
- Angles use 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.
Separate periodic signals by orthogonality
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Evaluate ∫₀^(2π) sin x sin 2x dx.
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
sin x sin 2x=[cos x−cos 3x]/2
The product-to-sum identity converts the mixed product into complete harmonics.
Work through the mathematics
∫₀^(2π)cos(kx)dx=0 for nonzero integer k
Each sine antiderivative returns to its starting value after the full period.
Check the conclusion
The integral is zero
Distinct sine modes are orthogonal under the full-period inner product, even though neither signal is identically zero.
The result
The integral is zero
Distinct sine modes are orthogonal under the full-period inner product, even though neither signal is identically zero.
Common mistakes to catch
- Orthogonal functions need not be zero at the same points.
- A normalization coefficient matters when extracting Fourier amplitudes.
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
Evaluate ∫₀^(2π)sin²x dx.
Show a hint
Use sin²x=(1−cos 2x)/2.
Reveal answer and explanation
π
The constant term contributes π and the oscillating term cancels.
Practice 2
Would a partial-period integral necessarily vanish?
Show a hint
Full-cycle cancellation may be lost.
Reveal answer and explanation
No
Orthogonality depends on the specified interval and inner product.
Take the idea with you
Extract one frequency component from an ideal periodic measurement.
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: Resolve four samples into discrete frequency components
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