Jan 21.pdf - Lecture Notes Winter 2019 MATA37 CALCULUS II FOR THE MATHEMATICAL SCIENCES LEC03 Jan 21st 2:00pm 3:00pm Instructor Email Office Office

Jan 21.pdf - Lecture Notes Winter 2019 MATA37 CALCULUS II...

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Lecture Notes Winter 2019 MATA37 - CALCULUS II FOR THE MATHEMATICAL SCIENCES LEC03 , Jan 21st, 2:00pm - 3:00pm Instructor: Dr. Kathleen Smith Email: [email protected] Office: IC458 Office Hours: TBA
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LEC03 , Jan 21st Richard Hong 2 INTEGRABILITY REFORMULATION 1 Recall Darboux definition of the definite integral, f is integrable on [a,b] ⇐⇒ sup { L ( f, P ) |∀ P [ a, b ] } = inf { U ( f, P ) |∀ P [ a, b ] } = Z b a f ( x ) dx 2 Integrability reformulation let [ a, b ] R , a < b, f is integrable on [a,b] ⇐⇒ ∀ > 0 , P partition of [ a, b ] such that U ( f, p ) - L ( f, p ) < Note that we do not compute definite integrals with this definition, only if the function is integrable. Example proof Consider g ( x ) = ( - 3 x Q 0 x / Q Prove Z 1 0 g ( x ) dx DNE by int reformulation. Show ¬ ( > 0 , P partition of [ a, b ] such that U ( f, p ) - L ( f, p ) < ) Show > 0 , 3 : P partition of [ a, b ] , U ( f, p ) - L ( f, p ) Choose > 0 Let P be an arbitrary partition of [0, 1] for i = 1 , 2 , 3 , . . . n , due to the density of Q , I any interval will contain both.
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