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3. Suppose g is a step function on [a, b). This means that there exists a partition a = yo < y1 < -.< YN = b and constants c so that g(r) = Ck for r E (yk-1, Yk). Show that g is integrable, with M g(x) dr =a(y% – Yk-1). k=1

3. Suppose g is a step function on [a, b). This means that there exists a partition a = yo < y1 < -.< YN = b and constants c so that g(r) = Ck for r E (yk-1, Yk). Show that g is integrable, with M g(x) dr =a(y% – Yk-1). k=1

Advanced Engineering Mathematics
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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**Text:** 3. Suppose \( g \) is a step function on \([a, b]\). This means that there exists a partition \( a = y_0 < y_1 < \cdots < y_N = b \) and constants \( c_k \) so that \( g(x) = c_k \) for \( x \in (y_{k-1}, y_k) \). Show that \( g \) is integrable, with \[ \int_a^b g(x) \, dx = \sum_{k=1}^N c_k (y_k - y_{k-1}). \] **Explanation:** This text discusses a step function \( g \) within the interval \([a, b]\). A step function is characterized by being constant over discrete intervals in its domain. The function \( g \) is defined as \( c_k \) over each segment \( (y_{k-1}, y_k) \), where \( a \) and \( b \) are endpoints of the partitioned interval divided into segments: \( a = y_0 < y_1 < \cdots < y_N = b \). The task is to prove that \( g \) is integrable over this interval and its integral equals: \[ \int_a^b g(x) \, dx = \sum_{k=1}^N c_k (y_k - y_{k-1}), \] which represents the sum of the areas of the rectangles formed by these constant values \( c_k \) across their corresponding intervals \( (y_{k-1}, y_k) \).
Transcribed Image Text:**Text:** 3. Suppose \( g \) is a step function on \([a, b]\). This means that there exists a partition \( a = y_0 < y_1 < \cdots < y_N = b \) and constants \( c_k \) so that \( g(x) = c_k \) for \( x \in (y_{k-1}, y_k) \). Show that \( g \) is integrable, with \[ \int_a^b g(x) \, dx = \sum_{k=1}^N c_k (y_k - y_{k-1}). \] **Explanation:** This text discusses a step function \( g \) within the interval \([a, b]\). A step function is characterized by being constant over discrete intervals in its domain. The function \( g \) is defined as \( c_k \) over each segment \( (y_{k-1}, y_k) \), where \( a \) and \( b \) are endpoints of the partitioned interval divided into segments: \( a = y_0 < y_1 < \cdots < y_N = b \). The task is to prove that \( g \) is integrable over this interval and its integral equals: \[ \int_a^b g(x) \, dx = \sum_{k=1}^N c_k (y_k - y_{k-1}), \] which represents the sum of the areas of the rectangles formed by these constant values \( c_k \) across their corresponding intervals \( (y_{k-1}, y_k) \).
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