Prove that da(x) = a(b) a(a), %3D |
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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note: second image is definition 2.1
![Definition 2.1 Let P
X0, X1, ..., Xn be a partition of [a, b] and let tk be a point in the
%3D
subinterval [xk-1, Xx]. A sum of the form
n
S(P f,a) - Σ ftu) Δαι
k=1
is called a Riemann-Stieltjes sum of f with respect to a. We say f is Riemann-integrable
with respect to a on [a, b], and we write "f E R(a) on [a, b]" if there exists a number A
having the following property: For every e > 0, there exists a partition Pe of [a, b] such that
for every partition P finer than Pe and for every choice of the points tr in [xk-1, xk], we have
|S(P, f,a) – A| .
When such a number A exists, it is uniquely determined and is denoted by f fda or by
L' f(x)dx . We also say that the Riemann-Stieltjes integral f fda exists. The functions f
and a are referred to as the integrand and the integrator, respectively. In the special case
when a(x) = x, we write S(P, f) instead of S(P, f, a), and f E R instead of f E R(a). The
integral is then called a Riemann integral and is denoted by fdx or by f(x)dx. The
numerical value of f" f(x)da(x) depends only on f,a, a, and b, and does not depend on the
symbol x. The letter x is a "dummy variable" and may be replaced by any other convenient
symbol.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff11bf13d-0137-4dc4-86b2-863ea249143f%2F7dcaa712-a832-4a06-b669-364f36caecb3%2Fkkiiuz_processed.png&w=3840&q=75)
Transcribed Image Text:Definition 2.1 Let P
X0, X1, ..., Xn be a partition of [a, b] and let tk be a point in the
%3D
subinterval [xk-1, Xx]. A sum of the form
n
S(P f,a) - Σ ftu) Δαι
k=1
is called a Riemann-Stieltjes sum of f with respect to a. We say f is Riemann-integrable
with respect to a on [a, b], and we write "f E R(a) on [a, b]" if there exists a number A
having the following property: For every e > 0, there exists a partition Pe of [a, b] such that
for every partition P finer than Pe and for every choice of the points tr in [xk-1, xk], we have
|S(P, f,a) – A| .
When such a number A exists, it is uniquely determined and is denoted by f fda or by
L' f(x)dx . We also say that the Riemann-Stieltjes integral f fda exists. The functions f
and a are referred to as the integrand and the integrator, respectively. In the special case
when a(x) = x, we write S(P, f) instead of S(P, f, a), and f E R instead of f E R(a). The
integral is then called a Riemann integral and is denoted by fdx or by f(x)dx. The
numerical value of f" f(x)da(x) depends only on f,a, a, and b, and does not depend on the
symbol x. The letter x is a "dummy variable" and may be replaced by any other convenient
symbol.
Transcribed Image Text:Prove that da(x) = «(b) – a(a), directly from Definition '2.1.
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