Let f [a, b] : → R be bounded and f (x) > a > 0, for all x € [a, b]. Show that if f is 1 Riemann integrable on [a, b] then 7 : [a, b] → R, (7)(x) = f() is also Riemann integrable on [a, b]. Hint: We show that given e > 0, there is a partition P of [a, b] such that U (7, P) – L (},P) < - < €. • Explain why we can find a partition P of [a, b] such that U (f, P) – L (ƒ, P) < a²€. • Show that M₂ (7) = m(f) and mi (}) = M₁(s). • Show that U (}, P) – L (7,P) ≤ & (U (ƒ, P) – L (ƒ,P)) < e.
Let f [a, b] : → R be bounded and f (x) > a > 0, for all x € [a, b]. Show that if f is 1 Riemann integrable on [a, b] then 7 : [a, b] → R, (7)(x) = f() is also Riemann integrable on [a, b]. Hint: We show that given e > 0, there is a partition P of [a, b] such that U (7, P) – L (},P) < - < €. • Explain why we can find a partition P of [a, b] such that U (f, P) – L (ƒ, P) < a²€. • Show that M₂ (7) = m(f) and mi (}) = M₁(s). • Show that U (}, P) – L (7,P) ≤ & (U (ƒ, P) – L (ƒ,P)) < e.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Please answer all parts clearly. Thanks
![Let f [a, b]
R be bounded and f(x) > a
7
0, for all x € [a, b]. Show that if ƒ is
is also Riemann integrable on [a, b].
1
Riemann integrable on [a, b] then : [a, b] → R, (3) (x)
f(x)
Hint: We show that given € > 0, there is a partition P of [a, b] such that U ( }, P) – L ( } , P) < e
● Explain why we can find a partition P of [a, b] such that U (ƒ, P) – L (ƒ, P) < a²e.
1
• Show that M₁ () = m() and m₁ (}) = M₁).
mi
• Show that U (, P) - L (7,P) ≤ 2 (U (f, P) – L (ƒ, P)) < e.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa8d46346-fe9f-45aa-a3a0-df12b7cae379%2F6feb41ee-d61a-4fae-96fe-8168f5405b7c%2Fbhx4wu_processed.png&w=3840&q=75)
Transcribed Image Text:Let f [a, b]
R be bounded and f(x) > a
7
0, for all x € [a, b]. Show that if ƒ is
is also Riemann integrable on [a, b].
1
Riemann integrable on [a, b] then : [a, b] → R, (3) (x)
f(x)
Hint: We show that given € > 0, there is a partition P of [a, b] such that U ( }, P) – L ( } , P) < e
● Explain why we can find a partition P of [a, b] such that U (ƒ, P) – L (ƒ, P) < a²e.
1
• Show that M₁ () = m() and m₁ (}) = M₁).
mi
• Show that U (, P) - L (7,P) ≤ 2 (U (f, P) – L (ƒ, P)) < e.
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