1. INTEGRALS (a) Let f: [a, b] →→ R be a Riemann integrable function and let c be a positive constant. Prove that cf is Riemann integrable and that •Sf(x)dr. (b) Let f: [a, b] → R be a Riemann integrable function. Prove that -f is Riemann integrable and that fe- f(x) dx = is continuous. = c. [₁-f(x) dx = f(x)dx. (c) Let f: [a, b] →→ R be a (bounded) Riemann integrable function. Show that the function g: [a, b] → R defined by g(x) = f* f a f(t)dt
1. INTEGRALS (a) Let f: [a, b] →→ R be a Riemann integrable function and let c be a positive constant. Prove that cf is Riemann integrable and that •Sf(x)dr. (b) Let f: [a, b] → R be a Riemann integrable function. Prove that -f is Riemann integrable and that fe- f(x) dx = is continuous. = c. [₁-f(x) dx = f(x)dx. (c) Let f: [a, b] →→ R be a (bounded) Riemann integrable function. Show that the function g: [a, b] → R defined by g(x) = f* f a f(t)dt
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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![1. INTEGRALS
(a) Let f: [a, b] →→ R be a Riemann integrable function and let c be a positive constant. Prove
that c-f is Riemann integrable and that
[ c. f(x)dx= = c.
is continuous.
e. [56
(b) Let f: [a, b] → R be a Riemann integrable function. Prove that -f is Riemann integrable
and that
ریم
f(x)dx.
-f(x)dr =- [* f(x)dx.
(c) Let f [a, b] → R be a (bounded) Riemann integrable function. Show that the function
g: [a, b] → R defined by
g(x) =
=ff(t)dt](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F04625ac1-43ff-4999-b93a-55388fc0c5e2%2F19ff7594-a605-4372-ae88-6bc2f3d0a19c%2F3bxhumk_processed.png&w=3840&q=75)
Transcribed Image Text:1. INTEGRALS
(a) Let f: [a, b] →→ R be a Riemann integrable function and let c be a positive constant. Prove
that c-f is Riemann integrable and that
[ c. f(x)dx= = c.
is continuous.
e. [56
(b) Let f: [a, b] → R be a Riemann integrable function. Prove that -f is Riemann integrable
and that
ریم
f(x)dx.
-f(x)dr =- [* f(x)dx.
(c) Let f [a, b] → R be a (bounded) Riemann integrable function. Show that the function
g: [a, b] → R defined by
g(x) =
=ff(t)dt
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