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Title: Riemann Integral - Definition
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Series: Real Analysis
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Chapter: Riemann Integral
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YouTube-Title: Real Analysis 51 | Riemann Integral - Definition
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Bright video: https://youtu.be/t8Hh73HxP1o
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Dark video: https://youtu.be/o-jagNl7kto
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Ad-free video: Watch Vimeo video
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Quiz: Test your knowledge
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Dark-PDF: Download PDF version of the dark video
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Print-PDF: Download printable PDF version
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Exercise Download PDF sheets
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Thumbnail (bright): Download PNG
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Thumbnail (dark): Download PNG
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Subtitle on GitHub: ra51_sub_eng.srt missing
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Timestamps (n/a)
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Subtitle in English (n/a)
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Quiz Content
Q1: What is the correct definition for a bounded function $f$ being Riemann-integrable?
A1: $$ \sup \left{ \int_a^b \phi(x) dx ~\bigg|~ \phi(x) \in \mathcal{S}( [a,b] ) , ~ \phi \leq f \right} = \inf \left{ \int_a^b \phi(x) dx ~\bigg|~ \phi(x) \in \mathcal{S}( [a,b] ) , ~ \phi \geq f \right} $$
A2: $$ \sup \left{ \int_a^b \phi(x) dx ~\bigg|~ \phi(x) \in \mathcal{S}( [a,b] ) , ~ \phi \geq f \right} = \inf \left{ \int_a^b \phi(x) dx ~\bigg|~ \phi(x) \in \mathcal{S}( [a,b] ) , ~ \phi \leq f \right} $$
A3: $$ \sup \left{ \int_a^b \phi(x) dx ~\bigg|~ \phi(x) \in \mathcal{S}( [a,b] ) , ~ \phi \geq f \right} < \inf \left{ \int_a^b \phi(x) dx ~\bigg|~ \phi(x) \in \mathcal{S}( [a,b] ) , ~ \phi < f \right} $$
A4: $$ \sup \left{ \int_a^b \phi(x) dx ~\bigg|~ \phi(x) \in \mathcal{S}( [a,b] ) , ~ \phi \geq f \right} > \inf \left{ \int_a^b \phi(x) dx ~\bigg|~ \phi(x) \in \mathcal{S}( [a,b] ) , ~ \phi > f \right} $$
Q2: Let $\psi : [0,2] \rightarrow \mathbb{R}$ be a step function. Is $\psi$ Riemann-integrable?
A1: Yes!
A2: No!
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Last update: 2025-01
