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(a) S (b) a (c) b (d) f(t) (e) dt (f) n (h) n-1 (i) f(t;) (j) Δt (k) lim 848 ned the standard 12.

(a) S (b) a (c) b (d) f(t) (e) dt (f) n (h) n-1 (i) f(t;) (j) Δt (k) lim 848 ned the standard 12.

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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## Understanding the Symbols in a Riemann Sum and Their Connection to Definite Integrals ### Mathematical Representation The definite integral of a function \( f(t) \) from \( a \) to \( b \) is given by: \[ \int_a^b f(t) \, dt = \lim_{n \to \infty} \sum_{i=0}^{n-1} f(t_i) \Delta t \] ### Explanation of Each Symbol #### (a) \(\int\) - **Purpose**: Represents the integral symbol, indicating the summation of infinite infinitesimal elements, \( f(t) \, dt \), over the interval \( [a, b] \). #### (b) \(a\) - **Purpose**: Lower limit of integration, representing the starting point of the interval on the real line. #### (c) \(b\) - **Purpose**: Upper limit of integration, representing the endpoint of the interval on the real line. #### (d) \(f(t)\) - **Purpose**: The function to be integrated, expressing the relationship between the variable \( t \) and the function's output. #### (e) \(dt\) - **Purpose**: Infinitesimal change in the variable \( t \), used to denote that integration is being performed with respect to \( t \). #### (f) \(n\) - **Purpose**: Number of subdivisions of the interval \([a, b]\), which approaches infinity in a Riemann sum. #### (g) \(i\) - **Purpose**: Index of summation, used to identify each subinterval and its corresponding evaluation point, \( t_i \). #### (h) \(n-1\) - **Purpose**: Shows that the sum includes all terms from 0 up to \( n-1 \), corresponding to the subdivisions of \([a, b]\). #### (i) \(f(t_i)\) - **Purpose**: Height of the rectangle at the subinterval associated with \( t_i \), where \( t_i \) is a sample point in the \( i \)-th subinterval. #### (j) \(\Delta t\) - **Purpose**: Width of each subinterval, calculated as \( \Delta t = \frac{b-a}{n} \). #### (k) \(\lim_{n \to
Transcribed Image Text:## Understanding the Symbols in a Riemann Sum and Their Connection to Definite Integrals ### Mathematical Representation The definite integral of a function \( f(t) \) from \( a \) to \( b \) is given by: \[ \int_a^b f(t) \, dt = \lim_{n \to \infty} \sum_{i=0}^{n-1} f(t_i) \Delta t \] ### Explanation of Each Symbol #### (a) \(\int\) - **Purpose**: Represents the integral symbol, indicating the summation of infinite infinitesimal elements, \( f(t) \, dt \), over the interval \( [a, b] \). #### (b) \(a\) - **Purpose**: Lower limit of integration, representing the starting point of the interval on the real line. #### (c) \(b\) - **Purpose**: Upper limit of integration, representing the endpoint of the interval on the real line. #### (d) \(f(t)\) - **Purpose**: The function to be integrated, expressing the relationship between the variable \( t \) and the function's output. #### (e) \(dt\) - **Purpose**: Infinitesimal change in the variable \( t \), used to denote that integration is being performed with respect to \( t \). #### (f) \(n\) - **Purpose**: Number of subdivisions of the interval \([a, b]\), which approaches infinity in a Riemann sum. #### (g) \(i\) - **Purpose**: Index of summation, used to identify each subinterval and its corresponding evaluation point, \( t_i \). #### (h) \(n-1\) - **Purpose**: Shows that the sum includes all terms from 0 up to \( n-1 \), corresponding to the subdivisions of \([a, b]\). #### (i) \(f(t_i)\) - **Purpose**: Height of the rectangle at the subinterval associated with \( t_i \), where \( t_i \) is a sample point in the \( i \)-th subinterval. #### (j) \(\Delta t\) - **Purpose**: Width of each subinterval, calculated as \( \Delta t = \frac{b-a}{n} \). #### (k) \(\lim_{n \to
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